Determination of the Differentiably Simple Rings with a Minimal Ideal

Annals of Mathematics
Determination of the Differentiably Simple Rings with a Minimal Ideal
Author(s): Richard E. Block
Source: The Annals of Mathematics, Second Series, Vol. 90, No. 3 (Nov., 1969), pp. 433-459
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1970745
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simple
Determination
of the differentiably
ringswitha minimalideal*
By RICHARD E. BLOCK
1.
In.troduc tionl.
A centralresultin ringtheoryis the Wedderburn-Artin
theoremon simple rings: a simpleassociative ring with DCC on left ideals is a total matrix
ringA.,over a divisionringA\.In this paper we consideran analogue of this
theoremin whichideals are replacedby differential
ideals, these being ideals
invariantunder all derivationsof the ring, and left ideals are replaced by
ideals. The main result is a completedeterminationof the differentiably
simpleringswitha minimalideal, in termsof the simple rings. This result
(theprecisestatementwillbe givenshortly)is neweven forfinite-dimensional
associative algebras with a unit. However, the result holds also for completelyarbitraryrings,not necessarilyassociativeand not necessarilyhaving
a unit element(just differentiably
simplewitha minimalideal). In the case
of Lie algebrasthe theoremprovesa thirty-year-old
conjectureof Zassenhaus.
In fact the resultleads to the solutionof importantproblemson two verydifferentclasses of finite-dimensional
non-associativealgebras, namely,semisimpleLie algebras (of characteristicp) and simple flexible(but not anticommutative)power-associativealgebras.
and notation. If A is a ringand if D is a set
We nowgive somedefinitions
of
A
of derivations
(additive mappingsd of A into A such that d(ab) =
(da)b + a(db) forall a, b in A) thenby a D-ideal of A is meantan ideal of A
invariantunderD. The ringis called D-simple (d-simpleif D consistsof a
single derivationd) if A2 # 0 and if A has no proper D-ideals. Also A is
called differentiably
simple if it is D-simpleforsome set of derivationsD of
A, and hence forthe set of all derivationsof A. The same definitionsare
used foralgebras over a ringK, the derivationsthenbeingassumedto be Klinear. Note that whenwe use the wordringor algebra we do notin general
assume,unless stated, that the ring or algebra is associative or has a unit
element; however,whenwe speak of an algebra over a ring,say over K, we
* This research was supportedby the National Science Foundation under grants GP
5949 through the University of Illinois, GP 6558 throughYale University,and GP 8713
throughthe Universityof California,Riverside.
RICHARD E. BLOCK
434
always assume that K is associative with a unit elementacting unitallyon
the algebra.
Jacobsonnoted(at least in a special case, see [16]) the followingclass of
simple rings A which are not simple: A is the
examples of differentiably
groupring SG whereS is a simpleringof primecharacteristicp and G # 1
is a finiteelementaryabelian p-group(so that G is a direct productof say n
copies (n > 1) of the cyclicgroupof orderp). If S is an algebraoverK then
SG is also an algebra over K. Since the ring or algebra SG depends (up to
onlyon S and n, we introducethe notationS[,,]forit. We only
isomorphism)
use this notationwhen S is simple of prime characteristic. We also let
S[,] denote S itself. If S is an algebra over K (so K= Z or ZP in the
ring case) then S[n] may also be writtenas S($? Bf,p(K) where Bn,P(K)
denotesthe (commutativeassociativewithunit)truncatedpolynomialalgebra
*I,. X]/(XP,
*. *,
XP) (p the characteristic of S).
Often one is inter-
ested in the case in which K is a fieldof characteristicp, when Bn,p(K) is
usually just writtenB,(K). Note that S[n] is associative or Lie etc. according as S is.
We now state the mostimportantresultof the paper.
MAIN THEOREM. If A is a differentiably simple ring (or K-algebra)
with a minimal ideal, theneitherA is simple or thereis a simple ring (or
K-algebra) S of prime characteristic and a positive integer n such that
A = S[n].
Converselyit is easy to see (and is a special case of results given below)
simpleand has a minimalideal,in facta unique
that each S[n]is differentiably
minimalideal. The assumptionthat A has a minimalideal cannotbe removed,
as is shownforexampleby polynomialand powerseries ringsover a fieldof
characteristic0.
simpleringmay be regardedas
Posner [11] notedthat any differentiably
centroid)and so has a characteristic.
an algebra over a field(its differential
Three special cases of the theoremabove were known. At characteristic0,
theseweredue to Posner
wherethe situationis comparativelyuncomplicated,
[11] (the associative case) and Sagle and Winter [15] (the case of finite-dimensionalalgebras). In the much more complicatedcharacteristicp case,
commutativeassociative
Harper [6] provedthe result for finite-dimensional
algebrasoveran algebraicallyclosedfieldF (theresultthenbeingA - RJF)),
whichproveda conjectureof Albert [3]. In the case of finite-dimensional
Lie algebras of characteristicp the problemwas also studied by Zassenhaus
[17] and Seligman[16]. Zassenhaus [17, p. 80] conjecturedthe resultin this
DIFFERENTIABLY
SIMPLE
RINGS
435
case in what we shall show in ?6 is an equivalentform.
simplerings is as menThe above resulton the non-simpledifferentiably
theorem. One aspect of that
tioned analogous to the Wedderburn-Artin
rings
in that theoremis played here
analogyis that the role of the division
by simplerings,and in the associative case divisionringsare preciselyanalogous to simple rings in that they may be characterizedas the non-trivial
ringswith no left ideals. Most of the proofof the Main Theorem(all of it
associativecases) is valid fora Din the characteristic0 or finite-dimensional
simpleringA (with minimalideal) when each operatord in D satisfiesa requirementweaker than that of being a derivation,namelyd is additiveand
[d, t] e T(A) for each t in T(A) where T(A) denotesthe ringgeneratedby all
by elementsof A. We call such an operatord
rightand leftmultiplications
a quasi-derivation.
The proofof the Main Theoremoccupies?2-?6. The followingis an outline of this proof. A key fact proved in ?2 is that A, as a module for its
factors. Also
serieswithisomorphic
algebra,has a composition
multiplication
A has a unique maximalideal N. In ?3 it is shownthat the centroidC of A
is also differentiably
simplewitha minimalideal,and that if C containsa subfieldE such that A/N is centralover E and if A (as an E algebra) containsa
subalgebraS -A/N with S + N = A then A S 0 EC; moreoverit is essentiallyshown that such an S exists if A is d-simpleforsomederivationd.
A proofof the commutativeassociative case is given in ?4, thus determining
C. In ?5 the d used to constructS is shownto exist by provingTheorem
5.2: if a ring A witha minimal ideal is D-simple whereD is a setof (quasi)derivationsclosed under addition, commutationand left multiplication by
all elementsof the centroidthenthereis a d in D such that A is d-simple.
simplering A with minimalideal
(As a veryspecial case, any differentiably
is d-simplefor some derivationd; when A = Bn(F) this gives a result of
Albert[3]). In ?6 the requiredfieldE is constructed,to completethe proof.
In ?7 we shall determinethe derivationalgebra der S[i] of S[o] in terms
of der S, and give a conditionon a set D of derivationsof S[,f]forS[,,]to be
D-simple.
In ?8 we shall considerdifferentiably
semisimplerings and give an anatheorem:a semisimpleassociative
logue of the followingWedderburn-Artin
ringwithDCC on leftideals is a directsum of simplerings. If D is a set of
(quasi)-derivationsof a ring A and if I is an ideal of A let ID denote the
(unique) largest D-ideal of A containedin I. We shall provein Theorem8.2
that if A has DCC on ideals and if A/I is a directsum of simple rings then
A/IL)is a directsum of rings whichare D-simple (or more preciselysimple
436
RICHARD E. BLOCK
with respectto the set of (quasi)-derivationsinducedon them by D). If A
is associative and R is the (Jacobson)radical of A, thenwe say that A is Dsemisimpleif RD = 0. As a corollary,if A is associative and D-semisimmple
with DCC on ideals and if AIR has DCC on left ideals, then A is a direct
sum of D-simple rings (and so is known,in termsof divisionrings). A similar result holds for alternativerings and a large class of finite-dimensional
power-associativerings. We shall also show that this gives a new proofand
extensionof the theoremof Oehmke [101 that semisimpleflexiblestrictly
power-associativealgebras are direct sums of simple algebras with a unit.
The authorhas announcedelsewhere[41 the determination,
by means of the
Main Theorem,of the symmetrized
algebra A+ forthe simpleflexiblealgebras
in the finite-dimensional
case. At the end of ?8 we shall state the generalization of this resultto the case of simple flexiblerings for which A+ has a
minimalideal.
For (finite-dimensional)
Lie algebras of primecharacteristicit is nottrue
that D-semisimplealgebras are direct sums of D-simple algebras since it is
not even true forsemisimplealgebras. Howeverin a semisim-ple
Lie algebra
L everyminimalideal I is (adL)-simple. This is the basis forthe application
in ?9 of the Main Theoremto obtaina structuretheoremforsemisimpleLie
algebras. The author hopes that ideas connectedwith the Main Theorem
will also proveusefulin the determination
of the simpleLie algebras.
2.
Chains of ideals
If A is an algebra over a ring K, we denote by T = T(A) the multiplication algebra of A, i.e., the (associative) subalgebra of HomK(A,A) (the
algebra of all K-linear additive mappings of A into A) generated by
y v-iyx,
{r,, lxi x e A} where rx and l1 are the right and leftmultiplications
s-of
A
do
not
1A
assume
e
T
the
that
y xy, respectively,
(1A
identity
(we
operator)). Also we let C = C(A) denotethe centroidof A, i.e., the centralizer
of T in HomK(A,A). If A2 = A then C is commutative(and associativewith
1). If D is a set of derivationsof A, the D-centroid of A (differential
centroidif D -der A (the derivationalgebra)) is definedto be the centralizer
of D in C. If A is D-simplethenthe D-centroidis a field;the proofis the
same as the usual one forthe centroidof a simplealgebra. We also notethat
if D is a set of derivationsof an algebra A over K thenA is D-simpleif and
onlyif A is D-simpleas a ring,again by the usual proofforthe case of ordinarysimplicity. Howeverfora suitable K thereexist K-algebraswhichare
differentiably
simpleas a ring but not as a K-algebra,as we shall see in ?7
(we requirethat derivationsof a K-algebrabe K-linear). In a D-simple K-
DIFFERENTIABLY SIMPLE RINGS
437
algebra A the set {x e A I T(A)x = 0} is a D-ideal and henceis 0. In particular
a minimalideal of A is an irreducibleT(A)-module. It followsthat a minimal
ideal of A as a K-algebra is also a minimalideal of A as a ring. Conversely
if A is botha D-simple ring and a K-algebra thena minimalideal of A as a
ringis also a (minimal)ideal of A as a K-algebra.
If d is a derivationof the algebra A then [d, rx] (= dr, - rxd) = rdx and
[d, lxj = ldx and hence [d, T] c T. We call a quasi-derivationof A any element d of HomK(A,A) such that [d, T] c T. Thus the set qderA of quasiderivationsof A formsthe Lie normalizerof T in HomK(A,A). A quasiderivationfor A as an algebra is also a quasi-derivationforA regardedas a
ringsince T is the same set forA regardedas an algebra or as a ring. The
followingare examples of quasi-derivationswhich are not in general derivations.
( 1 ) If A is associativeand I is an ideal, then (rxI I) e qderI for all x
in A.
( 2 ) If c e C(A) and d e derA, thend + c e qderA and dc e qderA (while
cd e derA). The above definitionsand facts (about the D-centroidetc.) go
over forquasi-derivationsas well.
LEMMA 2.1. Let H be an associative algebra overa ring K, let M be an
H-modulewith a minimal submoduleM1,and let D be a subsetof HomK(M,
M) such that [D, HM] C H, If M2,***,Mq (for someq > 1) are submodules
of Msuch that M1cM2c ***cMq and M,+1/M,-11M for i = 1, ..., q -1,
and if d e D such that dMq L Mq, then there is a submoduleMq?,, with
Mq c Mqi, and an index j, 1 < j < q, such that dMj_1c Mq (MO= 0),
dM1jc- Mq+i,and themapping m + Mj_1I- dm + Mq (m e Mj) is an isomorphism a of Mj/Mj-1ontoMq+l/Mq. In particular Mq+,/Mq Ml.
PROOF. If N is a submoduleof M then the mappingn v- dn + N of N
since dhn + N = hdn + [d, hM]n+ N(h C H).
into M/N is a homomorphism
Let j be the largest index (1 < j < q) such that dMyi c Mq, take N = Mq,
and considerthe restrictionto Mj of the above homomorphism.Let the
image be Mq+i/N; this definesMq+i. The kernelis Mj_1since M1is minimal
of Mj/Mj-1onto
and Mj/Mj-1_ Ml. This gives the requiredisomorphism
lemma's
is
and
the
Mq+i/Mq,
proof complete.
Our firstapplicationsof this lemmawill be with M = A (an algebra over
K), D a set of quasi-derivationsof A, and H= T(A), so that submodulesare
series of a ringor algebra,say of A, we
the same as ideals. By a composition
shall mean (unless otherwisestated) a compositionseries for A as a T(A)module,so that the termsof the series are ideals of A.
RICHARD E. BLOCK
438
simple algebra
LEMMA 2.2. Suppose that A is a quasi-differentiably
with a minimal ideal M1. Then A has a compositionseries and a unique
maximal ideal N. MoreoverN is nilpotent,A/N is simple, A2 = A, and
everycompositionfactor is isomorphicto M1as a T(A)-module.
series. Let d,.
PROOF. Supposethat A does nothave a composition
d...
be a (possiblyempty)sequenceofnot necessarilydistinctquasi-derivationsof
A. Let i1 be the firstindex(if any) such that di1M,X M1and use di1to define
an ideal M2as in Lemma2.1. If M2,***,Mr have already been constructedby
Lemma2.1 usingdi1, * * *, dir-1 wherei1 : ... < ir-1 and dir-i . . . d2d1Mj c Mr
let s = ir be the lowest index (if any) afterit such that d8Mr .7 Mr. Then
Lemma2.1 gives Mr,,. If dMr X Mr,,continueusing d, untilMr-, ..,* Mr+,
c
d2d1M1
...
Mr+u. Proceedinguntilthe indiceshave beenexhausted,we obtainM1, **,
with d. *. dM1 c M,. Then Mq # A since we are supposing that there is
no compositionseries,and thereis a quasi-derivationd such that dMq V Mq.
ApplyingLemma2.1 once morewe get an Mq+i. If m e M1and x = do . **d1m
then x e Mq and hence xM1 = M1x = 0 since Mq+I/Mq and M1 are T-isomorphic.
=
have been obtained such that dsMrC Mr+fu Then ir+fuf_
s
and d. *
e qderA, m e M1,n ? O0
But the subspace of A spanned by all do . ** d1m(dn
is closed under T and hence is A itself. ThereforeAM1 = M1A = 0. But
{a e A aA = Aa = O} is a quasi-differentialideal and hence A2 = 0, a
contradiction. Therefore A has a compositionseries, in fact has one
0 ci M1 ci * *
= A such that M-?1/M -M1
as
T-modules for
i
=
1,.*. *,lI- 1.
ideal
Let N= M1_. Then N is maximal. Since A2is a quasi-differential
(because we do not use 1A in generatingT), A2 = A and A/N is simple. Also
NM,+1 c Mi and M,+1N c M, for all i since NA C N and AN c N. Therefore every productof at least 21-1elementsof N, associated in any manner,
is zero. If N1 # N is a maximal ideal then N + N1 = A and A/N1 N/N f N,
is simpleand nilpotent,a contradiction.Hence N is unique. This completes
the proofof the lemma.
3. The centroidand a tensor factorization
If A is an algebra, d e qderA and c E C(A) then [d, c] e C(A) since if
t e T(A) then [[d, c], t] = [[d, t], c] + [d, [c, tII= 0 by the Jacobiidentityfor
commutators.It followsthat if d e qderA thenthemappingco-o[d,c](c e C(A))
is a derivationof C(A). We shall denotethis derivationby d*.
simple algebra over K. If
LEMMA 3.1. Let A be a quasi-differentiably
0
= Nc( M = A
A has a minimal ideal M1 and if = M ,CiM1C * CiMl1
DIFFERENTIABLY SIMPLE RINGS
439
is a compositionseries of A constructedas in Lemma 2.1 then there are
monomorphisms 1,,** , a, of C(A/N) into C(A), as K-modules, such that the
followingholdsfor i = 1, ***, 1. If 0 # v E C(A/N) thenoi(y)A + Mi-1 = Mi,
i()N (--Mi-,, and if 8 is any T(A)-isomorphismof A/N onto Mi/Mi-1then
there exists a fi in C(A/N) such that the mapping of A/N into Mi/Mi-,
inducedby ai(RS)is 8.
PROOF. By Lemma 2.1 thereis a T-moduleisomorphism
p of A/N onto
M1. Writer= C(A/N) and definea, by v1(y)=eipwryw (y e f) where wi denotesthe naturalprojectionof A ontoA/Miand pi is the injectionof Mi into
and 1(y): A > A, so that a,
A. The factorsof r1(y)are T-homomorphisms
maps F into C(A). If 0 # Y E r theny is ontoand o1(7)A = M1,and hencea,
of A/N onto M1 then p-10c F, (p-10)is one-one. If 0 is a T-isomorphism
p1Ozcl-,,and p-10is the requiredA.
In the construction of M2, ***, Ml by Lemma 2.1, let di denote the quasi-
indexand 6t
derivationused to go fromMi to Mi+1,withji the corresponding
onto
of
the correspondingT-isomorphism Mj,/Mj,_1
Mi+1/Mi. We define
by setting oi,1() -[di, ri,(7)]. Thus ui, maps F into
(a, recursively
2, *
C(A). Suppose the conclusions hold for r < i (i < 1). If y # 0, then
dijuj(y)A+ Mi = Mi+, and ajo(7)diA c Mi. Hence oi+1(7)A+ Mi = Mi, and
ui+,is one-one(and clearlyK-linear). Similarlyui+1(7)NC Mi. If 0 is a Tof A/N onto
of A/N onto Mi?l/Mi,then bi10is a T-isomorphism
isomorphism
Mj./Mj,-,. Hence there is a 8 in r such that ajc(,3) induces t`0, i.e.,
Vii-laii(IS)= i-10v,2(where we regardaj as a mappingof A intoMj and ijis the projectionof Mj onto Mj/Mj-1). Then ui+,(,S)= [di, j,(r1)]induces 0
since viujr(fl)di 0 and b6ij,-,= vidion Mji. This completesthe proof.
LEMMA 3.2. Let A be a D-simple algebra overK with a minimal ideal
M1,whereD c qderA. Then C(A) is D*-simple,whereD* denotes{d* I d E D}
(and d*(c) = [d, c] (c e C(A))). MoreoverC(A) has a compositionseries, with
thesame lengthI as that of A.
PROOF. ( i ) If c e C = C(A) and cA c Mi (i > 0) (we use the notation
is a T-homoof Lemma 3.1) thencN c Mi-1: Assumingthat cA 4 Mi-1,7ri-1c
morphismof A intoA/Mi-1with image Mi/Mi_1;the kernelis a maximalideal
of A, so by Lemma 2.2 is N.
(ii) C =
ai(r) (where 7 = C(A/N)): It sufficesto show that if
0
cA c Mi, cA Mi-, thenthereexists a S in r such that (c - a(,8))A c Mi-1.
0 of A/N onto Mi/Mi-1. By
Since cN c Mi-1, c induces a T-isomorphism
Lemma 3.1 there is a 8 in r such that ai(S) also induces0, and this 8 has
the requiredproperty.
440
RICHARD E. BLOCK
(iii) I = {c C C IcA c M1} is a minimal ideal of C: It is obviouslyan
ideal. To show that it is minimalwe show that if c, c' e I, c # 0 # c', then
there exists c" C C such that c' = cc". Here c and c' induce T-isomorphisms
8 and 8' of A/N onto M1,and the requiredc" is an element(which exists by
O-'O'of A/N.
Lemma 3.1) whichinducesthe T-automorphism
(iv) C is D*-simple: Suppose that H is a non-zeroD*-ideal of C. Then
HA = {I hjaj} is D-closed since d(ha) = h(da) + (d*h)a e HA forall d in D.
Hence HA = A, and so thereis an h in H such that hA + N = A. Then if
o # v e F we have 0 # u1(y)he H n I. Since H is D*-closed,(ii) and the constructionof the cr in Lemma 3.1 show that C -E ui(F) c H, and H = C.
-{ e C I cA c Mj0,
.*, ,form a com>,(r)
(v) The ideals E.=
positionseries of C: The equality followsfromthe proofof (ii). The expressionon the right side shows that theyare ideals, C being commutative.
The expressionon the left side and the constructionof the ai in the proofof
Lemma 3.1 show that the ideals are obtainedby the methodof Lemma 2.1
startingfromthe minimalideal I, and thus forma compositionseries of C.
This completesthe proofof the lemma.
LEMMA3.3. Let A, D and Ml be as in Lemma 3.2, and let N be the
unique maximal ideal of A. Suppose that E is a subfieldof C(A) such that
A/N as an E-algebra is central. If r is any mapping of A/N into A which
NA
A/N -0 regarded as a sequence of
splits the exact sequence 0
E-module homomorphisms,then the mapping sO co-oc(rs) (s e S = A/N,
c e C(A)) gives an isomorphismz', as C-modules,of S 0 EC onto A, and C
has dimension 1 overE. If r can be chosenpreservingproducts(that is, if
the above sequencesplits whenregardedas consistingof algebra homomorphisms) thena' is a (C, E and K)-algebra isomorphism.
PROOF. Since C/(radicalC) is a field,E acts unitallyon A and A is an Ealgebra. Since CN c N, S = A/N is also an E-algebra. Let ct = ag(1s),i =
1, *., 1 (we continue using the notation of Lemma 3.1). Then co, *-- cl are
linearlyindependentover E, since if c = c; + E<j ccejthen cA + Mj-, = Mj
(because cA + M-, = M,) and c # 0. If cA c M, and cA ? M,1 thenc and
E and F-, of A/N onto M,/Mj_,. Then (c)-'U
ci both induce T-isomorphisms,
is a T-automorphism
of A/N, hence is in the centroidof S as a K-algebra.
Thereforethere is an e in E such that wi1(c - eci) = 0. It follows that
c,, ..,
c, are a basis of C over E.
For a given splitting E-homomorphism
v, (s, c) v->c(vs) is E-bilinear,
hence v' is well-defined.Every elementof S ( EC is uniquelyrepresentedas
Ei== Si 80co,si C S. If 0 # E. si 0 ci C kerv', let j be the largest indexsuch
DIFFERENTIABLY SIMPLE RINGS
441
that z-sj# 0. Since z-sjX N and cj induces a T-isomorphism
of A/N onto
MJ/M>_1,
and kerz' = 0.
cjrsj i Mj-1. But A <j cirsiC M>1,, a contradiction,
Obviouslyz' is a homomorphism
of C-modules,wherec'(s (0 c) = s (0 c'c. If
T preserves products then cc'z(s1s2)
=
(czs1)(c'vs2), so that r' is an isomorphism
of algebras (over C, hencealso over E and K).
the lemma.
This completesthe proofof
3.4. Suppose that A is a d-simple algebra, where d is a deriS,( A/N)
No A
m
0 splits
(as a sequenceof algebra homomorphisms),
and in fact S' ={a GA Ida G M1}
is a splittingsubalgebraz-S.
LEMMA
vation, with a minimal ideal M1. Then 0
Since d is a derivationS' is a subalgebra. The chain of ideals
Mg ...9, Ml-1, N, Ml = A may be constructedwith d always used to go
fromMi to Mi,1 (so that ji = i). If 0 # a e S' f N, let i be the indexsuch
thata C Mi, a X Mi-1. Thenda X Mi by Lemma2.1, whichcontradictsdS' c M1.
ThereforeS' n N 0
O. MoreoverdN + M1 = A since it contains with Mi
also dMi +Mi = Mi-1. Hence if aGA then da =dn + a1 where nGN and
a, e M1,d(a - n) = a1,a = (a - n) + n C S' + N. This completesthe proof
of the lemma.
PROOF.
4. The commutativeassociative case
Suppose that A is a differentiably
simple commutativeassociative ring.
At characteristic0, resultsof Posner [11] show that A is an integraldomain.
In particularif A has a minimalideal thenA is a field. For the application
to the proofof the Main Theoremit wouldsufficeto provethisassumingthat
the radical N of A is nilpotent,A/N is a field,and A has a unit element;the
l
proof in this case is trivial: If y C N, yi # 0, yi+s
0 and dy X N for some
derivationd, thendy is a unitbut yi(dy)= d(y`+1)/(i
+ 1) = 0, a contradiction.
Now supposethat A has primecharacteristic.Harper [6] provedthat if A is
a finite-dimensional
algebra over a fieldE and if A has the formA = El + N
(N theradical),thenA B-(E)
forsomen > 0. The followingresultgeneralizes Harper's theoremin several ways. A portionof the proof is partly
based on Harper's proof,but yieldsa shorterproofeven of his special case.
For the ringsconsidered,the differential
constants(i.e. elementsannihilated
by all derivations)forma subfieldwhichmaybe identified
withthedifferential
centroidundera restrictionof the mappingx +- lx which identifiesthe ring
withits centroid.
THEOREM4.1. Let A be a differentiably
simple commutativeassociative
ring of prime characteristicp, and let R ={x C A IxP = O}. If Rx = 0 for
RICHARD E. BLOCK
442
some x # 0 in A (this will hold,e.g., if A has a minimal ideal), thenthere
is a subfieldE of A and an n > 0 such thatA BJ(E) (- EJ), in fact isomorphicas algebras overE. Here E may be taken to be any maximal subfield of A containingthe subfieldF of differentialconstants.'
PROOF. By two theoremsof Posner [11J,A has a unit elementand the
ideal R is nilpotent(for the applicationto the proofof the Main Theoremit
would have been enoughjust to assume these properties). By Zorn's lemma
thereexists a maximalsubfieldof A containingF; let E be any suchmaximal
subfield. We now give the proofof the theoremin fiveparts.
( i ) E + R = A: If a C A then al e F, say aP = a, and the minimalI
polynomialof a over E divides XP- a. If E containsno pth rootof a, then
xp- a is irreducibleover E and E(a) is a field,which contradictsthe maximality of E. Therefore E contains a pth root 8 of a, a - S e R, and
aGEE- R.
Now regardA as an algebra over E, choosea subset {yj Ij C J} of R such
that {ly + R'f I C J} is a basis of R/R2,let X = {X Ii C J} be a corresponding
over E, and let z be the (unique) homomorset of commutingindeterminates
phismof E[X] to A such that z-Xj = yj(j C J) and zi -1. Then
(ii) z is onto: It suffices(since R is nilpotent)to show for all k > 0
that
i+
m+ Rkl--r|
J, iI > 0,
ijf k}
*+
{ ii .*
jrliX
This is true fork 1 by the choiceof {yj}, and its truth for
spans Rk/Rk-l-.
k + 1 followsfromthat fork because
RkR
A5 (EyJi ...
myj
f)
+ Rk~l)(Eyi + Rd
C
(Ey;~ Y*
...
Yi + Rk).
Now regardA again as a ring.
(iii) Given d in derA, there exists an F-linear derivation d' of E[XI
such that zd' = dz: Let {wi} be a basis of A over E composedof monomials
in the yi's. If we write de = A, (die)wi(ee E, die e E) then each d, is a
derivationof E and foreach e in E, die = 0 except fora finiteset of indices
i. For each wi let zi be the elementof E[XJ obtainedby replacingthe y/'s
by Xi's. Also foreach j in J pick an Xj' in E[X] such that tX;. dy,. Then
thereis a (unique) F-linear derivationd' of E[X] such that d'X, = X( j E J)
and d'(el)
(die)zi(ee E), as a straightforwardverificationshows, and
this is the requiredd'.
'Added April 25, 1969. The writer has just learned of the paper by Shuen Yuan, Differentiablysimple rings of prime characteristic,Duke Math. J. 31 (1964).623-630,in which
essentiallythe result of Theorem4.1 is proved (under the hypothesisthat the radical is
nilpotent). The proof given here is differentand simpler than Yuan's, and also makes
the proof of the Main Theoremself-contained.
DIFFERENTIABLY SIMPLE RINGS
443
}id ederA zd' = dz}; then ker z is a
(iv) Let D' ={d'ederFE[XI
D'-ideal and a derFE[X]-ideal of E[X]: kerz is obviouslya D'-ideal
Amaximal
simple. Let I be the additive
and is maximalby (iii) since A is differentiably
subgroupofE[X] generatedby kerz + A(kerz) whereA = derFE[XI. Then
I is an ideal because b(6k) = 3(bk)- (b)k (b e E[XI, 3 e A, k e kerZ), and I is
D'-closed because d'(3k) = 3(d'k) + [d', 3Jkand [d', 61]E A. By maximality,
either I = kern and hence A(kerz) c kerI, or else I = E[XI, in which case
kerz wouldcontainan elementwitha non-zero(monomial)termof degree< 1.
But this latter contradictsthe fact that {1, yj Ij e J} is linearlyindependent
moduloR' and z maps monomialsof degree > 1 intoRi.
( v ) kerz = (XP), where(XP) denotestheideal generatedby {XP Ij E J1
and theindex set J is finite: (XP) c kerT. But (XI) is a maximal A-ideal of
E[X] (just apply enoughderivationsa/Dxjto an elementnot in (XP) to get a
unitmodulo(XP)). Hence kerz = (XP), and the nilpotencyof R impliesthat
J is finite. This completesthe proofof the theorem.
simple commutativeassociativering
COROLLARY4.2. If a differentiably
A of prime characteristichas ACC on nilpotentideals, then A Bn(E) for
somefield E and some(finite)n > 0.
PROOF. With {yj j e J} as in the proofof the theorem,if J' c J is finite
then the ideal generated by {yj j e J'} is nilpotent. Hence J is finite,say
ideal.
J n. Then,as in (ii), RP' - RP"t, and RP=(R=)2 is a differential
Hence R is nilpotent,and the resultfollowsfromthe theorem.
5. Proof of d-simplicity
forma Lie algebra
If A is anyalgebra overa ringK, the quasi-derivations
(the Lie normalizerof T(A)), as do the derivations.
overK undercommutation
If c e C(A) and d e qderA thencd e qderA, since [cd,t] -[c, t]d + c[d,t] and
c7T(A)c T(A) (cl = laC.,
crx - rz. If d e derA then it is easy to see that
but notnecessarilya derivation). Thus
ed C derA (and dc is a quasi-derivation
qderA is a left C(A)-moduleand derA is a submodule.
LEMMA 5.1. Let A, D, and D* be as in Lemma 3.2. If C(A) is d*-simple
for a given d in D thenA is d-simple. If D is closed under commutation
and leftmultiplicationbyelementsof C(A) thenD* is also.
PROOF. Suppose that C = C(A) is d*-simpleand that M + 0 is a d-ideal
of A. Since A has a T(A)-composition
series,M containsa minimalideal A1
of A. Startingwith M, constructthe chain of ideals M, by always using the
givend in Lemma2.1 untilan ideal Mq is obtainedwith dMQ cI M,. If Mq + A
then H = {c e C I cA c Mq}is a properideal of C, but if h e H and a e A then
444
RICHARD E. BLOCK
(d*h)a = dha - hda C Mq, so that H is d*-closed,a contradiction. Therefore
Mq= A, M = A and A is d-simple. The last statement holds because
[ad d1,ad d2]= ad[d,, d2]in Hom (A, A), and c(d*c,) = cdc - ccd = (cd)*c,.
If A is a ringor algebra and if D is a Lie subringor subalgebraof qderA,
we say that D is regular if it is also a left C(A)-submoduleof qderA. This
conformswith the terminologyused by Ree [12] in his investigationof the
regular Lie subalgebrasof derBJ(F). If A is an algebra and if A2 = A then
the centroidof A is the same set whetherA is regardedas an algebraor as a
ring. Hence a regularLie subringof qder ,A is also a regular Lie subringof
qderA whenA is regardedas a ring.
THEOREM5.2. If A is a D-simple algebra overK with a minimal ideal
where D is a regular Lie subring of qderA thenA is d-simplefor somed
in D.
PROOF. Since A is D-simple(d-simple)as an algebra if and only if it is
D-simple(d-simple)as a ring (d E D c qderKA),we may ignoreK and regard
A as a ring (the minimalideal remainsminimal). Also we mayassumethatA
is not simple. By Lemmas 5.1 and 3.2 we may assume thatA is commutative
associativeand that D c derA. Then by Lemma 2.2 and Theorem4.1, A has
primecharacteristicp and A- B(E) forsome fieldE and some n > 0. We
claim that
( i ) there is a regular Lie subring Do of D and a derivationd, in D
such thatA is notDA-simple,A is (Do U {d1})-simple,and [di,Do] c Do: Since
the radical R of A is not a D-ideal, thereis an r in R and a d, in D such that
dlr e R and hencedir is a unitof A. Replacingd1by (dlr)-'d,we mayassume
1 Let Do- {d - (dr)dl1d E D}. Then Do {d E D I dr = O} and
that dlr I.
hence Do is a regular Lie subringof D, and [d1,DO] c Do since if dor= 0
then (dldo - dod,)r= - dt1 0. Also A is not DO-simplesince rA is a proper
DO-ideal. Any(Do U {d1})-idealof A is invariantunderd - (dr)dl and (dr)dl for
all d in D, and hence A is (Do U {d1})-simple.
Now let H be the (associative) subringof Hom(A, A) generatedby T(A)
and Do. Then [d1H] c H and A has no properH-submoduleinvariantunder
di. Since H-submodulesare in particularideals of A, A has an H-composition
series. Startingwith a minimalH-submoduleM1 of A and applying Lemma
2.1 usingd, at each step, we get an H-compositionseries 0 = MoC ***c Ml =
= Mj/Mj-, (with
A and H-isomorphisms ti: Mi
Mi, where we write AMj
Ml = A). Since M, is a DO-idealof A, any d in Do inducesa derivationon A
(regardedas a ring)whichwe denote by d. We also write D, = {d I d C Do}*
Then Do is a regularLie ringof derivationsof A, and A is DO-simple.By in-
DIFFERENTIABLY
445
RINGS
SIMPLE
ductionon the (T(A)) compositionlength(M, # 0 since A is not DO-simple)
thereis a doin Do such that A is do-simple. We have
(ii) if A is not d1-simplethenA Bq(E) for somepositiveq: By Theorem4.1, A Bq(E) forsome q > 0 withthe same fieldE as above (this latter fact is not actually needed) since A/N A/N. Since d1 gives all the
mappingsQ,,if 0 # a e A thenforsome i, dia X M11. But if q 0 thendia is
a unit since A is a fieldand M,1 is nil, and hence any non-zerod1-idealwould
containa unit.
Now suppose q > 0, let x1,*--, xq be a set of nilpotentgeneratorsof A
(i.e., A = E[x1, *
Xi = O. i _ 1, *--, q; E a copy of E), and for i=
1, ..., I1let Mi, M,' denotethe ideals of A such that M, D M' D M _'D
6
* * *, Xq), and M1'/M1,._- (61o1...*
= (elk **,)'l(x1,
'/AM,_
)-'E(x, ... xq)p
to the unique
i.e., M'/Mi-,and MI'/Mi, correspondunderthe H-isomorphism
maximaland minimalideals of A. Also pick yi in A such that yi + M1, = xi
(i=
1, *.,q) and set w = (Y1 * yq)p1. Then
(iii) wM' c M$'1 (whereMt- 0) and wMi + Mi 1 = M,'(i = 1, *
,
1):
We have
d1w + Ml-, =
(p - 1)(d1y, ? Ml
*
.
so that each term in d1w+ Ml-1 is of total degree at least (p -_ )q - 1.
Ml + M,1, (dlw)M' c M', and, by the
+
Hence (d1w+
Ml1-)(M' Ml1-) c
1, ..., I1). Also wMi + Mi1 = M"' for
H-isomorphisms,(d1w)M' 5 M''(i
of M' and M'', d1M' + Mi =
all i sincethisholdsfori 1. By the definition
Also wM'
0 since
M'.1 and d1M'' + M- M!'1 (i = 1, *--,I-1).
Suppose wM' c M'1 forsome i (1 < i < 1; this is truefori =
wM' c
Ml-,.
[d1,I<] = ldjw)
Then
(since
1).
wM'_ 5 wd1M' + wM, c d1(wM') + (d1w)M' + M'' c d1Mll1 + M"' c Ml' .
(iv) Let d = do+ wd1; then A is d-simple: In provingthis, for any
ideal I of A we writeAI and AOIforthe ideal dI + I and doI + I, respectively,
and has the same
and similarlyfor a0 on ideals of A. Since A is do-simple
compositionlengthas its dimensionover E, it followsthat
(AO) 2-2(R(xl ...
Xq)V-()
X1, *.
,
Xq)
and hence
(A)Pq-2M1l'
= M(=1,.**,I).
Therefore,since d1M' 5 M'.1, (iii) gives AiM'' = AMm, ApqM'' = AM=
, pq-1; i = 1, ,I). But for i < I, AM=-wd1M + M,=
M, (j = 1,
jM7, j - 1, ..., (pq-1)1,
toMl' 1. Therefore the ideals
Mll
+
wM,1
getherwith the 0 ideal, forma compositionseries of A. Hence A is d-simple
446
RICHARD E. BLOCK
since M"' is the unique minimalideal of A (or by the argumentof (ii)). This
completesthe proofof the theorem.
6. Completionof proof of the main theorem
Let A be a differentiably
simple algebra over K with a minimalideal
A.
Since
der
A
a
is
regular
Lie ring,by Theorem5.2 thereis a derivaM, #
tion d such that A is d-simple. By Lemma 3.2, C = C(A) is d*-simpleand
I = {c c C I cA C M1}is a minimalideal of C. By Lemma 2.2, the radical R of
C is nilpotent,and by Lemma 3.4, E ={c E C Id*c e I} is a subfieldof C with
E + R -C. The differential
constantsof C are in E and in particularK4 c E.
By ?4, C (and hence also A) has primecharacteristicand C BJ(E), as an
algebra over E, for some n > 0 (R # 0 since otherwiseC E -I and A =
M1). Hence I is 1-dimensional
over E.
The ringA is also an algebra over E. In the notationof Lemma 3.1, if
0 vy P -C(A/N) thena1(Y)A M1,hence CM1 (- M1,and M, is an E-ideal.
Since [d, E]A = (d*E) c MJ,in the constructionof the chain of ideals Mi
using d in Lemma 2.1, all the isomorphisms
a are E-linear; in particularA/N
and P may be consideredas algebras over E. Thereforethe mappingp used
in definingul in Lemma 3.1 (U,(r) = pepawrl)is E-linear,and q, is an isomorphismof P ontoI as E-modules. ThereforeA/N as an E-algebra is central.
(la e A I da C M1} of Lemma 3.4 is closed underE, so we
The subalgebra S'
have the situationof Lemma 3.3, and A S' 0 EBfl(E) S't1]as an algebra
over E, hencealso over K, where S' A/N is a simple algebra over E and
hencealso over K. This completesthe proofof the Main Theorem.2
One of the main tools used in the proofof the Main Theorem was the
passage fromquasi-derivationsof A to derivationsof C(A). The proof of
Lemma 3.4 was the only place wherewe required,forthe D-simplealgebra
A itself,that D consistof derivationsrather than merelyquasi-derivations.
The followinggives several cases wherethis hypothesismay be weakened.
6.1. Let A be a D-simple algebra with a minimal ideal,
COROLLARY
whereD c qderA. Then theconclusionof theMain Theoremholds in each
of thefollowingcases
( i ) A has characteristic0;
(ii) A is commutativeassociative with a unit element;
(iii) A is finite-dimensionalovera field and is either alternative (including associative), or standard [1], [14] (including Jordan)of character2AddedApril 25, 1969. The proof of the Main Theorem remains valid for Lie triple
systems,and more generally for Q-algebras A where A is a K-module and each weQ is
n-ary multilinearfor some n _ 2 (with the appropriatedefinitionsof derivation,etc.).
DIFFERENTIABLY
SIMPLE
RINGS
447
istic # 2;
(iv) D = {d} and for each x, y in A there is a t = t(x,y) in T(A) such
that d(xy) - (dx)y + t(dy).
PROOF. Case (i) followsfromLemma 3.2 and the (easy) characteristic0
case of ? 4. Case (ii) followsfromLemma 3.2 and the Main Theorem since
here A- C(A). Case (iii) followsfromthe Wedderburnprincipaltheorem
for alternativealgebras [13] and for standard algebras [14] of characteristic # 2, since this providesthe splittingsubalgebraneededin Theorem3.3,
thus bypassingLemma 3.4 (and ? 5). In Case (iv) the set S' of the proofof
Lemma3.4 is a subalgebra,and so as notedabove the conclusionholds in this
case also. This concludesthe proof.
The Main Theoremprobablyremains true for any quasi-differentiably
simplealgebra witha minimalideal. At any rate, Lemma 3.3 and Theorem
4.1, togetherwith Theorem5.2 and the argumentof ?6, providea great deal
of information
about such an algebra. It can also be seen that in the use we
of quasi-derivationcould be
have made so far of the concept,the definition
furtherweakened.
Givenone of the non-simplealgebrasA of the Main Theorem,thealgebra
since it is the uniquesimpledifS is uniquelydeterminedup to isomorphism,
ferencealgebra of A, and n is uniquelydetermined
sincepf is the composition
lengthof A(p the characteristicof S and of A). One of the principaldifficulties in provingthe Main Theoremwas findinga subalgebraof A corresponding
to S. It is easy to see that for any two such subalgebras there will be an
of A sendingone to the other. Howeverthe subalgebrais not
automorphism
in generalunique. To show this we need onlygive an automorphism
of A =
S (0 ZPB(Zp) which moves S 0 1. Thus suppose that S has a derivation
d' 0 and that x e Bn(Zp) with x # x2 -0, and let d be the linearmapping
of A determinedby d(s 0 b) = d'(s) 0 xb (s e S, b e B,(Zp)). Then d is a derivation of A, d # d2= 0, exp d = 1A + d is an automorphismof A, and
(exp d)(S ? 1) # S Q 1.
We now discuss the conjectureof Zassenhausmentionedin ? 1. For any
Lie algebra S overa fieldF of characteristicp and any integerm > 0 he constructed[17,pp. 64, 67] a Lie algebra S(m',whichhe called a powerringof S.
This consists of the vector space over F of all mn-tuplesa = (a,,
elementsof S, withproductsgiven by [a, b] = c where
, a.-,) of
(i = 0,1, *--. m-1)
.j0(')[ajj bi-j]
Based on his characterizationof the d-simpleLie algebrasoverF witha chief
series[17,pp. 61-79],Zassenhausconjectured[17,p. 80]
(= T(A)-composition)
Gt =
448
RICHARD E. BLOCK
that if L is a differentiably
simpleLie algebra over F (presumablywitha
chief series) then L -Slv" for some simpleLie algebra S over F and some
n > 0. It turnsout that we can show rathereasily (we omitthedetails)that
S(P%)and S[f,]= S X FBfl(F) are isomorphic,underthe followingmapping
(SS *s, p)
-
pT-i!-si
'
>
? x{l
...
an
+ ip"-', 0 < ij < p (i = 0, *.,p
denotes
-1),
where i = i, + i2p +
and Bn(F) = F[x1, *--, xJ, xn = 0
the residue class modulo p of l/pordPl,
(j = 1, *- , n). Zassenhaus' conjecture then follows immediately from the
Main Theorem. It seems clear that the tensorproductformof Sn] used in
the presentpaper is a muchmoreconvenientconstructionthan that of the
powerrings.
7. The derivations of S[ff]and a condition for D-simplicity
In view of the Main Theorem,it is desirableto determinethe derivations
of the algebra S[n] of that theorem. This we shall now do, of coursein terms
the derivationsof a large
of the derivationsof S. We beginby determining
class of tensorproducts.
7.1. Let A = S ( FB, regardedas an algebra over F, where
F is a field,S and B are algebras overF, and B is commutativeassociative
with a unit element. If S or B is finite-dimensionaland if S2 = S or
THEOREM
{sGSIsS
= Ss =O} 0 O then
der A = (derS) (
FB + 1
0
F(der B)
(thesum beinga directsum as vectorspaces) where r denotesthe centroid
of S and
(d' ( b1)(s0 b) = (d's) (0 (bob),
(- (0 d")(s (0 b) = (-ys)0 (d"b)
(d' C derS, d" C derB)
verification
showsthateach d' 0 b and X0d"
PROOF. A straightforward
is a derivationof A. Now let d be a given derivationof A. We give the
proofin threeparts.
( i ) Proof of thetheoremwhenS is associativewitha unit element. We
identifythe centerof S with P. Choose a basis {biI i C I} of B and a basis
{Isj j C J} of F, and extend the latter to a basis {is Ij C J'} of S where the
indexset J' = J U J" with J flJ" 0. Definelinear transformations
d' on
S (i GI) and do'on B (j G J') by writing
d(s 0 1) =
(d's) (0 b
,
d(1 (0 b) =
Hje
s0
sj
J,
(df'b)
(seS, beB)
449
DIFFERENTIABLY SIMPLE RINGS
Then each d' is a derivationof S since
0 bi= (d(s,0
I
d'(s8s2)
0 1) + (s,0 1)d(S20 1)
+ s,(d's2)}0 bi
= S1{(dXs1)s2
1))(S2
and similarly each do' is a derivation of S.
Applying d to both sides of
(s 1)(1 0 b) = (10b)(s ( 1) gives (s 0 1)d(l X b) = d(l X b)(s 1) since
1 0 b commutes with d(s 0 1), and hence 0 -Ej,
(d;'b)=
[s, sj]
for
all
s
in
and
b
in
B.
For
a
S
b
in
B
and
each j
given
[8, sj] 0 (d!b)
LjeJ.
in J" write dCb= Lie aijbi (ay e F). Then
0
=
je
[s, sj]
0
(dCb) =
Ael [s, Eject aijsj3]0&bi
forall s. Hence ei J,' ajsj G F forall i, aj = 0 forall i and all j in J", and
d;! = 0 forall j in J". Then
d(s (0 b)
d((s (0 1)(1 0 b)) = Ad (d's) 0
+
bib
=.(ds)
bib + Ejs sj d;'b
0 d;'b
,
(seS, beB) s
S
and d has the formA. d' 0 bi + Ej,
sj 0&d7. By the definitionof the d',
foreach s in S, dXs= 0 except forfinitely
manyof the indicesi. Thereforeif
is
d'
0
S
then
finite-dimensional,
except for finitelymany i, so that
E d, (0 bie (derS) 0 B, while of coursethe same is true if I is finite. Similarly Eie sj A) d;! e F 0 (derB). Finally, the sum is direct since if
0derB thenE, (d' 0 bi)(s0 1) 0 for all s, and d- 0 for
E, d 0 bie FN
all i. This completesthe proofwhen S is associativewith 1.
(ii ) If U = P + T(S) thenthereis an algebra isomorphismz of U0(&B
ontoC(A) + T(A) with z(u 0&b) _U 0 lb (regardedas a mappingof A, where
lb is themultiplicationbyb on B), and F and C(A) are commutative. U is
an (associative)algebraof lineartransformations
on S since71,=-1, and yr8
rrs. There is a unique linearmappingz of U0 B ontoa space of lineartransformationsof A with z(u 0&b) = 0 lb (U e U, b e B). Thenz is one-onesince
if (L. ui 0 b4)(s0 1) = 0 forall s thenui = 0 for all i. Clearly z preserves
multiplication. Therefore z(T(S) ( B)= T(A) since T(ls 0 b) = 1b and
7(r.0 b) = r,8?&b.If yr,
I' e r then [y,7y]S2 =([/1 7']S)S = S(7, 7']S) = 0 and
hence the hypothesison S impliesthat P is commutative.Also eitherA2 A
or {a e A I aA = Aa =} = 0, so that C(A) is commutative. We claim that
(0 B) = C(A). Indeed if c e C(A) and we write c(s 0&1) = Ad y(s) (0 bi
(J(r
(ieS), we see that each yieF. But if beB then 11S0lbe C(A) and so
c(s & b) = (15 0 lb)C(S 0 1) = go -$(S) & bib,and c = Ad 7i bi. Here 7i - O
except for finitelymany i, as in (i). Therefore z7(F0 B) = C(A) and
T(UO(&B)
=
C(A) + T(A).
450
RICHARD E. BLOCK
(iii) Proof of the theorem when S is not associative or does not have a
unit element. Since [d, T(A)] c T(A) and [d, C(A)]
C(A), commutation by
d gives a derivation of C(A) + T(A), and hence by (ii) also a derivation d of
U 0) B. The center of U is F since F is commutative. Since U is associative
with a unit element, if {ya Ij C J} is a basis of F and {bi I i C I} of B then by
(i) there are derivations d' of U and d;' of B such that
ld(s(@b)
[d, ls b] =Zd(l8
(& b)
=zTA. GId~ti(l8)0@
bib + z
jl0 9 d;!b
Then with b = 1 and with d'(i e I) definedby setting d(s 01)
(so that as in (i) each d$ is a derivation of S) we have
T
Hence Odo~s) = d
'(l,)
ldi
s)
Lir
9
l
=
7
L)
i
beB)
(seS,
0
d'(s)
=
b-
(s
1
C
S)
forall i, and (since 7jl=, 1
l0d(s(3b)-,?Eidi(s)(@bib-A
jgjrjs(@djbI
-
(s
?
e
S,
b C B)
and the same equalityholdswithl replacedby r. Thendo d d 0 bi -jC <y5j?) digis a derivationof A, 0 = (doS)S = S(doS) doS2, and do 0.
The desired conclusionfollowsas in (i). This completesthe proof of the
theorem.
In the courseof the proofwe have also establishedthe followingresult
forthe infinite-dimensional
case.
COROLLARY7.2. Let the hypotheses of the first sentence of Theorem 7.1
hold. If S has a unit element, then every derivation d of A has the form
,d' ( bi + Ej 7i (9 do' where d' E der S, d.' CderB, and for each s in S
(resp. b in B), d'(s) = 0 (resp. d;(b) = 0) except for finitely many i (resp. j)
(and every mapping of this form is a derivation).
In orderto dropthe hypothesisthat S2 -- S or {s e S I sS = Ss =O} 0=
we would have to take into account summands
d"'S
{sS
I sS = Ss = 0}.
d"'
where d"'S2
0 and
of derSr,,]when S is a
We now apply Theorem7.1 to the determination
simplealgebra over a ring K and n > 0. Let E be the centroidof S, so that
E is a field containing K5, and identify A = S[r] with S 0) EBf(E), with S
identifiedwith S 0&1. If b e BR(E) then the E-linear mapping Zb of S 0&EBf(E)
determined by Zb(S 0) b') = s 0 bb' (s e S, b' e BR(E)) is in C(A). It is easy to
show (see [4]) that every element of C(A) has this form. If [d, Zb] = 0 for all
derivations of the form d = 0s(9 d" (d" e der EBfl(E)) then d"b = 0 for all d",
and b e El. Hence the differentialcentroid F of A is a subfield of E, the
latter being regardedhereas acting on A (and of courseK4 c: F). In particu-
451
DIFFERENTIABLY SIMPLE RINGS
lar S is also an F-algebra, and we may now identifyA with S ?FBfl(F) and
der A with der,(S 0FBfl(F)). If d' e derS then d'(01 e der(S 0&PB.(Z,))
and hence d' is F-linear. Thus derS -derFS. Also derFB"(F) is the well
Jacobson-Wittalgebra We,over F. We have now
known npn-dimensional
provedthe followingresult.
COROLLARY7.3. If S is a simple algebra (of prime characteristic)over
a ring K and if n > 0 then derS[},]= (derS) (&FBfl(F) + F OF Wn(regarded
as acting on S OFBfl(F)), whereF is the differentialcentroidof Sr,], r is
thecentroidof S, and W,.is theJacobson-Wittalgebra overF.
We now give a conditionon a set D of derivationsof SEn,]for S[,] to be
D-simple. If A = S 0&EB,where S is simplewith centroidE and B = B(E)
and if d =
d' 0 bi Ji
+
(0 d" e derA, we shall say thatd" is the component
of d in derB, and denote it by dB. A straightforwardcomputationshows
that ZdBb - [d, Zb1 forall b in B.
PROPOSITION 7.4. Let A = S[q,], identifiedwith S 0) EBJ(E), where S is
a simple algebra over a ring K and E = C(S), and let D be a set of E-linear
derivations of A. Then A is D-simple if and only if B = Bn(E) is DBsimple,
simple,whereDB -{dB I d e D}. In particular, S[%1is differentiably
and in fact d-simplefor someE-linear derivationd.
PROOF. In provingthe firstconclusionwe may ignore K and regard S
and A as algebras over E. We firstshow that everyideal of A has the form
S 0) H where H is an ideal of B. Thus let M be an ideal of A. If
Li sj f0 bj C M where the sj are linearlyindependentover E, then, by the
density theoremapplied to T(S), for each j there is an s' in S such that
s' bj cM. If s &bcMthen S b C M. Itfollowsthat{bcB13scSwith
s be M} is an ideal H of B, and M = S 0&H. If H is any DB-idealof B
thenS 0&H is a D-ideal of A. Converselyif S C) H is a D-ideal we see that
dBH C H forall d in D. It is easy to see that an ideal H of B is a DB-ideal
if and onlyif S 0&H is a D-ideal of A. This gives the firstconclusion. Now
let d = is 0 d" whered" is the derivationof B given by
+ xPj(Da/x2) +
d"r= (D/Dx1)
...
+ (xi ...X.-..
-lax"')
.
n)). Then B
j,1 x? = 0, (y(a/xi)xj = 6jy(i, j = 1,
is d"-simple[3], d is K-and E-linear,and A is d-simple(just to show that A
n}). This
is differentiably
simple,it sufficesto use {is 0&D/DxI i = 1, * , n
completesthe proof.
In discussingthe converseof the Main Theoremit remainsto show that
PBn(ZP),where
SE,,,](S simple)has a minimalideal. IdentifySn] with S
(where B = E[x1,
...,
452
RICHARD E. BLOCK
Then it is easy to see that
**., n).
ZAx1,*.*, I x.], X4 0 (i
S ?9 (x1
xX)-1 is a minimalideal of S[,, and in fact is containedin every
non-zeroideal of S[.].
The followingis an exampleof an algebra A over a ring K such that A
simpleas a ring but not as an algebra over K. Let S be a
is differentiably
) for some n > 0, K
central simple algebra over Z,, A = S (0ZB(Z
Bn(Zp),and regardA as a K-algebra in the obvious way. Then a derivation
of A, as a ring,is K-linearif and onlyif it has the formE, d' 0 bi and A, as
ideal.
a K-algebra,has S 0 (xl, *.*, x) as a properdifferential
B.(Z)=
8. D-semisimplerings
In this sectionwe obtaina D-structuretheoremwhichgives an analogue
theoremwhich says that a semisimple
of the part of the Wedderburn-Artin
artinianringis a directsum of simplerings. We state the results forrings;
theycan be extendedto the case of algebras withoutdifficulty.
Let A be a ringand D a set of quasi-derivationsof A. If I is an ideal of
A thenthe sum of all D-ideals of A containedin I is a D-ideal of A whichwe
denote by ID. If I is any D-ideal of A then D induces a set DA/1of quasiderivationsof A/I, and by a slight imprecisionin language we shall speak of
D-ideals of A/I ratherthan DA/[-ideals.Similarlyif D consistsof derivations
of A then D induces a set of derivationson I, and if D consistsof quasiderivationsand I is a direct summandof A then D inducesa set of quasiderivationson I; in eithercase, if I is (D II)-simple we shall also say that I
is D-simple.
If the ideal I is an intersectionf M,of ideals of A thenit is easy to see
f
If A is associative and R is the (Jacobson)radical of A
that ID n(Mk)D
then we call RD the D-radical of A. For alternativerings(whichof course
includeall associative rings)the radicalR is again takento be the intersection
of the regularmaximalleft ideals, and RD is taken to be the D-radical. Then
again R equals the intersectionof the primitiveideals [7] and also is the
largest ideal I whichis radical in the sense that foreveryx in I the leftideal
generatedby {yx - y y e A} containsx (and so in the associative case every
x has a quasi-inverse). Thus for alternativeringsas well as forassociative
rings we have two characterizationsof the D-radical: RD is the (unique)
largest radical D-ideal, and RD flPD where the intersectionis over all
Lie algebra then the
primitiveideals P of A. If A is a finite-dimensional
radical R is taken to be the (unique) largest solvable ideal, and in a context
in whichfinite-dimensional
algebras are being discussedthe
power-associative
radical R is taken to be the (unique) largest nil ideal (this agrees with the
DIFFERENTIABLY SIMPLE RINGS
453
in the alternativecase). In all these cases we call RD the
previousdefinition
0
D-radical of A and we say that A is D-semisimpleif RD
O. It is easy to see
that A/RD is D-semisimple.
We shall use the followingdual to Lemma 2.1.
LEMMA 8.1. Let H be an associative ring, let M be an H-module with
a maximal submoduleM1,and let D be a subset of Hom(M,M) such that
J
[D, HM]
Hm. If M2, ***, Mq (for some q > 1) are submodules of M such
that M1D M2z ... D Mq and M1/Mj+1 M/M1for i =1,1 ., q -1, and if
d e D such that dXJIq
t Mq, thenthereis a submoduleMq,+,with MqD Mq+j,
and an indexj, 1 _ j _ q, such thatdMq c Mj_1(Mo= M), dMq+lc Mj, and
themapping m + Mq,+H-> dm + Mj (m e Mq) is an isomorphismof Mq/Mq+
ontoMj..Mj. In particular Mq/Mq+ MIM,.
PROOF. Let j be the largest index (1 < j < q) such that dMq c Mj-,
and let Mq+= {m e Mq Idme Mj}. As in the proof of Lemma 2.1, the restrictionto Mq of the mappingm E-+dm + Mj is a homomorphism
with image
Mj-1/Mjand kernelMq,+,and thus gives the requiredisomorphism.
THEOREM8.2. Let I be an ideal of a ring A, let D bea setof quasi-derivations of A, and suppose that A/ID has DCC on ideals. If A/I is a direct
sum of simple rings then A/ID is a direct sum of D-simple rings. In fact
*.*
if A/I = Si
Sk (Si simple) then A/ID
S1G1( ...
* & SkGk where,
for each i, Gi = 1 or Si has prime characteristicpi and Gi is a finiteelementaryabelian pi-group.
PROOF. Withoutloss of generalitywe may assume that ID = 0. We first
supposethat A/I is simple,and apply Lemma8.1 with H = T(A), M = A and
M1 = I. By DCC the chain of ideals constructedby Lemma 8.1 cannotbe extended past some ideal Ml. Then Ml is a D-ideal and henceMl = 0. Since
(A/I)2 = A/I, T(A)(M1/Mj+1) = Mj1Mj+1 for i = 0, 1, ..., 1 - 1, T(A)A = A,
and A2 = A. Also I is nilpotent,as in Lemma 2.2. Since A has a T(A)-compositionseries, any D-ideal L # A is containedin a maximal ideal I'. If
I' zL I thenI + I' = A and A/I'-I/I
=
A2= A.
n I' is nilpotent,contradicting
ThereforeI' = I D L, L = 0, and A is D-simple.
Next supposethat A/I = (L1/I) E ... e* (Lk/I) wherethe Lj are ideals of
A containingI such that Lj/I = Sj is simple(k is necessarilyfiniteby DCc).
For j = 1, ...* k, let P3 = Eioj Li. Then A/Pj (A/j)/(Pj/I) _ Sj is simple, and A/PjD is D-simple with unique maximal ideal Pj. Moreover
fl
= ID. We may choosean index l < k and
fnPkD=
PlDfn
nPk)D
(P1 n
a reordering of the Pj such that PiD n ... n PID = ID and such that if
z of
1 < i < 1 thenMj = n i=1, ,l;ioj PiD # ID. Considerthe homomorphism
e
e
454
RICHARD E. BLOCK
A into (A!PD) (D ...*
(AIPJ) given by zx = (x + PlD,* X + PlD)(x E A).
This has kernelID. Since M3! PjD, PjD + M
A. Hence forany x + PjD,
there is a y such that yA = (0, .., x + PjD, .., 0). Then z is onto and
A/ID- (A/PD) GD... & (AlP1)). Countingthe numberof maximal ideals on
both sides we get 1 > k, and so l = k. This and the Main Theoremcomplete
the proof.
We remarkthat the theoremwould be false withoutthe assumptionof
DCC on ideals, as is shownby the followingexample. A = F[xJ (F a fieldof
characteristic0), I= (1 + x), D ={x(a/lx)}, whereID 0 but (x) is a D-ideal.
COROLLARY8.3. Let A be an (associative or) alternativering with DCC
on ideals and with radical R, and let D be a set of quasi-derivationson A.
If A is D-semisimpleand AIR has DCC on left ideals thenA is a directsum
of D-simple rings.
PROOF. A/R is a subdirectsum of primitiverings which are either associativeor Cayleyrings[7]. Since AIR is artinianit is a finitedirectsum of
simpleringsby the same argumentas in the associativecase. Hence Theorem
8.2 gives the result.
We now discuss a similarresult for an importantclass of power-associative rings,the flexiblerings. A ringA is calledflexibleif (xy)x = x(yx) for
all x, y in A. Oehmke[10] provedthata finite-dimensional
semisimpleflexible
strictlypower-associativealgebra over a fieldof characteristic# 2, 3 is a
direct sum of simple algebras with a unit element. Using Theorem8.2 we
now give a new proofof this result,extendingit to includethe characteristic
3 case as well as generalizingit to a resulton D-semisimplealgebras. In givand facts. Let A be
ing this resultwe make use of the followingdefinitions
an algebra over a fieldof characteristic# 2. Then A+ denotesthe algebra
x *y =
vectorspace as A and withnewmultiplication
withthe same underlying
(1/2)(xy+ yx). For each x in A let dxbe the mappingof A into A definedby
shows that each dx is a derivation
dxy= [x, y] (y e A). A directverification
of A+ if (and onlyif) A is flexible. Obviouslya subspace of A is an ideal of
A if and onlyif it is a {dxI x E A}-idealof A+.
COROLLARY8.4. Let A be a finite-dimensional
power-associativealgebra
overa fieldof characteristic# 2, with nilradical R and with a given (possibly empty)set D of quasi-derivations,and supposethatA is D-semisimple.
If A/R is flexibleand strictlypower-associativethenA is directsum of Dsimple algebras (and is flexible,strictlypower-associativeand has a unit
element). If A has a traceform then again A is a directsum of D-simple
Jordanalgebra).
algebras (and at characteristic# 5 is a non-commutative
DIFFERENTIABLY
SIMPLE
RINGS
455
PROOF. The trace formcase followsimmediatlyfromTheorem8.2 and
Albert's theoremon trace forms[13,p. 136]. Also whenAIR is commutative
the result holds by Theorem 8.2 and the fact [2, 8] that semisimplecommutativestrictlypower-associativealgebras are directsums of simplealgebras witha unit element(if S has unit elementthenso does SG) (the strictness of the power-associativity
is relevantonly at characteristics3 and 5).
Now supposethat A/Ris flexibleand strictlypower-associative.Then (A/R)+
is commutativeand strictlypower-associative.Also (A/R)+ has no non-zero
nil {d Ix E A/R}-ideal. By the just proved commutativecase, (A/R)+ is a
direct sum of {dxI x e A/R}-simpleideals which have a unit element. These
ideals of (A/R)+are also simpleideals of AIR, and the unit elementof (A/R)+
is also the unit element of AIR; this latter fact followsquicklyfromthe
froman idempotentdeflexiblelaw, as in [4], or, using power-associativity,
composition.We may now apply Theorem8.2 to get the desiredresult. This
completesthe proof.
The decompositionof a D-semisimplering intoD-simpleideals can also
associative case by using the minibe provedeasily in the finite-dimensional
mal D-ideals. Before discoveringthe present version of Theorem 8.2 the
authorhad used the minimalrather than maximal ideals to give a proofof
Theorem8.2 in the case when A is a finite-dimensional
power-associative
algebra and the Si have a unit element. This also gave Corollaries8.3 and
8.4. This was announcedbrieflyin [5]. After[5] was submitted,the author
receivedfromT. S. Ravisankar,of Madras, India, a copyof a recentmanuscript which gives a proof,differentfromthe author's, that if a finite-dimensionalflexiblestrictlypower-associativealgebra of characteristic# 2, 3
is D-semisimple(D a set of derivations)thenit satisfiesthe conclusionof the
firstpart of Corollary8.4. He also establishes in another proof the trace
formcase (again excludingcharacteristic3).
Whilediscussingsemisimpleflexiblealgebras we also mentionthe deeper
the structureof A+ for a simpleflexiblealgebra A
questionof determining
(of characteristic# 2). This was answered by the author [4] in the finitedimensionalcase as an applicationof the Main Theoremof the presentpaper.
Aside fromgivinga uniformproofof the various special cases proved by a
numberof authors (see [4]), this solved the previouslyunsettleddegree2
characteristicp case, includedas well considerationof nil simplealgebras,
and showedthat the resultdoes not dependon power-associativity.Sincethe
Main Theoremhas now been proved in a more general formthan when [41
was written,we now have the followingformof the resulton simpleflexible
rings.
RICHARD E. BLOCK
456
THEOREM8.5. Let A be a simpleflexiblering of characteristic# 2. If
and if A+ has a minimal ideal, theneitherA+ is
A is not anti-commutative
simple or the characteristic is prime and A+-B- (E) for somen > 0 and
somefield E containingthecentroidof A.
case in [4].
The proofis essentiallythe same as forthe finite-dimensional
A+
is
the
Main
Theorem
that
if
A+ is not
Since
differentiably
simple,
implies
simplethenthe characteristicis primeand thereis a commutativesimplering
forsome n > 0. The problemthen is to show that S
S such that A+ - S[Ef]
must be associative, and this may be done by showingthat S = El where
E = C(S); forthe details of the proofof this, see [41.
9. Semisimple Lie algebras
In this final section we apply the Main Theorem to the study of the
structureof finitedimensionalsemisimpleLie algebras of characteristicp.
The reason that this can be done is that in a Lie algebra L a non-abelian
ideal M is minimalif and onlyif M is (adL)-simple. We begin with a preliminaryresult on the relation between L and its minimalideals. All the
Lie algebrasconsideredare assumedto be finite-dimensional
over a field. For
any Lie algebra L we writeinderL forthe Lie algebra of innerderivations
of L, i.e., inderL = {ad x I x E L}.
9.1. Let D be a set of derivations of a finite-dimensionalLie
algebra L and suppose that L is D-semisimple. Then L has onlyfinitely
LEMMA
many minimal D-ideals, say L1, ***, Li, their sum M is direct, each Li is
(D U inderL)-simple, themapping x v--adMx(xE L) is an isomorphismof L
ontoa subalgebraof derM containing inderM, and the mapping d v--d IM
(d e D) of D into derM is one-one.
PROOF. Without loss of generalitywe may assume that D is a subalgebra of derL containinginder L. Since L is centerlesswe may identify
L with inderL and thus since [d, adLx] adL(dx) we may assume that L is
an ideal of D with 0 centralizerin D. Then it followsthat the D-ideals of L
are exactlythe ideals of D containedin L, and D is semisimple. Therefore
by replacingL by D it will sufficeto provethe resultwhen L is semi-simple
and D = inderL. With theseassumptions,let M = L, + *- - + Lr be a maximal directsum of minimalideals of L. Then M containseveryminimalideal
of L, the annihilatorin L of M containsno minimalideal of L (since any
such would be abelian) and so is 0, and hence x e->ad~x is an isomorphismof
L intoderM. The remainingconclusionsfolloweasily. This completesthe
proof.
DIFFERENTIABLY SIMPLE RINGS
457
In the case in whichD - derL, the resultsof Lemma 9.1, exceptforthe
finalstatement,are due to Seligman[16],but the proofgiven here is shorter.
COROLLARY9.2. Let L be a finite-dimensionalsemisimple Lie algebra
with a set D of derivations. Then any minimal D-ideal of L is a minimal
ideal of L.
PROOF. A minimalD-ideal is differentiably
simpleand so any ideal of L
properlycontainedin it is nilpotentand hence 0.
We call the sum of the minimalideals of a Lie algebra L the socle of L,
and if D is a set of derivationsof L we call the sum of the minimalD-ideals
of L the D-socle of L. Thus if L is semisimplethe D-socle equals the socle.
The determinationof the differentiably
simplealgebras and their derivationalgebrasleads quicklyvia Lemm9.1 to the followingdescriptionof all
the semisimpleLie algebras (and theirderivations)in termsof the socle.
Let S1, *--, S. be simple Lie algebras over a field F of characteristic p,
and let n1, *--, nr be non-negative integers (not necessarily distinct). Write
> Si 0 Bi whereBi denotesBn.(F) and Bi = F if ni = 0 (all algebras
S =
and tensorproductsconsideredhere are overF), and identifyS with inderS,
so that
(inderSi) (0 Bi
c derS er1 ((derSi) 0 Bi + Pi 0 derBi)
S = inderS =
whereFi denotesthe centroidof Si. Now let L be any subalgebra of derS
containing S (hence L is uniquely determined by a subalgebra of
1, ** , r let Li denotethe set
e3r=1((outder Si) 0 Bi + Fi 0 der Bj)). For i
of componentsin der(Si 0 Bi) of elementsof L, and if S is central,so that
der(Si 0 Bi) = (derSi) 0 Bi + 1,. 0 derBi, let LB. denote the set of comof ?7).
ponentsin derBi of elementsof Li (in the terminology
9.3. Every finite-dimensional semisimple Lie algebra of
prime characteristicis isomorphicto one of thealgebras L just constructed,
and the semisimple algebra uniquely determinesr and the pairs (Si, ni),
i
1, . *, r, up to isomorphismand reordering. The algebraL constructed
is semisimpleif and only if Si 0&Bi is Li-simple (which when Si is central
THEOREM
is equivalent to Bi being LBi-simple) for i = 1, *- -, r. The mapping
x e->ad1lx(x e N,,, L) is an isomorphismof thenormalizerof L in derS onto
derL. The same results hold with differentiablysemisimple in place of
semisimple provided the conditionon Li (or LB.) is replaced by the same
condition on
PROOF.
(Ndr
sL)i(or(NdersL)Bi)-
By Lemma 9.1 any semisimplealgebra withsocleMis isomorphic
RICHARD E. BLOCK
458
to a subalgebraof derM containinginderM, and by the Main TheoremM is
isomorphicto some S of the above formforsuitablepairs (Si, ni), thesebeing
essentiallyuniquelydeterminedby M. This gives the firstsentence. The
ideal Si 0DBi of L is minimalif and onlyif it is L-simple,or equivalentlyL,simple, and when Si is central this is equivalent to B. being LB-simple.
Since the centralizerof S in L is 0, every minimalideal of L is containedin
S. Hence if each Si ?gBi is minimalthen L is semisimple. Converselyif L
is semisimplethen each Si 0&Bi is minimalbecause any ideal of L properly
containedin Si 0&B. would also be a properideal of Si ?&Bi and hence nilpotent. This proves the second sentence. The mappingz definedby z-x
into derL. If adLx 0 then ad~x = 0
adLx(xe N. LL) is a homomorphism
and x = 0. Hence z is one-one. Suppose d E derL and let x be the unique
element of derS such that x(s) = [x, s] = d(s) (s E S). If y C L and s e S
then
(dy)(s) - [dy,s] = -[y, ds] + d[y,s] = -ly, [x, s]J+ [x, [y,s]j = Ix, y](s) .
Hence d = ad LX, x E Nler, 8L and z is onto. The finalstatementmaybe proved
the same as the firsttwo by replacingsemisimple,socle, ideal by differentiably semisimple,etc., and L-simple by (der L)-simple,using the characterization of derL just obtained. This completesthe proof.
COROLLARY9.4. Let F be a field of characteristicp. If every simple
Lie algebra over F of dimension< m has all its derivations inner then
Lie algebra over F of dimension< min{m + 1, 3p} is a
everysemnisimple
direct sum of simple algebras.
This is an immediateconsequenceof Theorem9.3, and gives a generalizationof Kostrikin'sresult[9] that a semisimpleLie algebra of dimension< p
over an algebraicallyclosed fieldof characteristicp > 5 is a direct sum of
simplealgebras (of classical type). On the other hand, Theorem9.3 shows
centralsimpleLie algebra over F, we can
that startingfroma 3-dimensional
constructa semisimpleLie algebra of dimension3p + 1 and a perfectsemisimpleLie algebra of dimension4p, neitherof whichis a directsumof simple
algebras.
UNIVERSITY
OF CALIFORNIA, RIVERSIDE
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