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 Accessed: 07/11/2010 19:03
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http://www.jstor.org Determinationof the differentiablysimple ringswith a minimalideal*
By RICHARD E. BLOCK
1. In.troduc tionl. A centralresult in ring theoryis the Wedderburn-Artintheorem on sim- ple rings: a simpleassociative ring with DCC on left ideals is a total matrix ringA., over a divisionring A\. In this paper we consideran analogue of this theoremin which ideals are replacedby differentialideals, these being ideals invariantunder all derivationsof the ring, and left ideals are replaced by ideals. The main result is a completedetermination of the differentiably simplerings with a minimalideal, in termsof the simple rings. This result (the precisestatement will be givenshortly) is new even forfinite-dimensional associative algebras with a unit. However, the result holds also for com- pletelyarbitrary rings, not necessarilyassociative and not necessarilyhaving a unit element(just differentiablysimple with a minimalideal). In the case of Lie algebras the theoremproves a thirty-year-oldconjecture of Zassenhaus. In fact the resultleads to the solutionof importantproblems on two verydif- ferentclasses of finite-dimensionalnon-associative algebras, namely,semi- simple Lie algebras (of characteristicp) and simple flexible(but not anti- commutative)power-associative algebras. We now give somedefinitions and notation. If A is a ringand if D is a set of derivationsof A (additive mappings d 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 invariantunder D. The ring is 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 differentiablysimple if it is D-simplefor some set of derivationsD of A, and hence forthe set of all derivationsof A. The same definitionsare used foralgebras over a ring K, the derivationsthen being assumed to be K- linear. Note that when we use the wordring or algebra we do not in general assume, unless stated, that the ring or algebra is associative or has a unit element; however,when we speak of an algebra over a ring,say over K, we
* This research was supported by the National Science Foundation under grants GP 5949 through the University of Illinois, GP 6558 throughYale University,and GP 8713 throughthe Universityof California,Riverside. 434 RICHARD E. BLOCK 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 examples of differentiablysimple rings A which are not simple: A is the group ring SG whereS is a simplering of primecharacteristic p and G # 1 is a finiteelementary abelian p-group(so that G is a direct productof say n copies (n > 1) of the cyclicgroup of orderp). If S is an algebra overK then SG is also an algebra over K. Since the ring or algebra SG depends (up to isomorphism)only on S and n, we introducethe notationS[,,] for it. We only use this notation when 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 (commutativeassociative with unit) truncatedpolynomial algebra *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. accord- ing as S is. We now state the most importantresult of the paper.
MAIN THEOREM. If A is a differentiably simple ring (or K-algebra) with a minimal ideal, theneither A 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) that each S[n]is differentiablysimple and has a minimalideal, in fact a unique minimalideal. The assumptionthat A has a minimalideal cannotbe removed, as is shownfor example by polynomialand powerseries ringsover a fieldof characteristic0. Posner [11] notedthat any differentiablysimple ring may be regardedas an algebra over a field(its differentialcentroid) and so has a characteristic. Three special cases of the theoremabove were known. At characteristic0, wherethe situationis comparativelyuncomplicated, these were due to Posner [11] (the associative case) and Sagle and Winter [15] (the case of finite-di- mensionalalgebras). In the much more complicatedcharacteristic p case, Harper [6] proved the result for finite-dimensionalcommutative associative algebrasover an algebraicallyclosed field F (the resultthen being A - RJF)), whichproved a 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 result in this DIFFERENTIABLY SIMPLE RINGS 435 case in what we shall show in ?6 is an equivalentform. The above resulton the non-simpledifferentiably simple rings is as men- tioned analogous to the Wedderburn-Artintheorem. One aspect of that analogy is that the role of the divisionrings in that theoremis played here by simplerings, and in the associative case divisionrings are preciselyanalo- gous 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 in the characteristic0 or finite-dimensionalassociative cases) is valid fora D- simplering A (with minimalideal) when each operatord in D satisfiesa re- quirementweaker than that of being a derivation,namely d is additiveand [d, t] e T(A) for each t in T(A) where T(A) denotesthe ringgenerated by all rightand left multiplicationsby elementsof A. We call such an operatord a quasi-derivation. The proofof the Main Theoremoccupies ?2-?6. The followingis an out- line of this proof. A key fact proved in ?2 is that A, as a module for its multiplicationalgebra, has a compositionseries with isomorphic factors. Also A has a unique maximalideal N. In ?3 it is shownthat the centroidC of A is also differentiablysimple with a minimalideal, and that if C containsa sub- fieldE 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 es- sentiallyshown that such an S exists if A is d-simplefor some derivationd. 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 set of (quasi)- derivationsclosed under addition, commutationand left multiplication by all elementsof the centroidthen there is a d in D such that A is d-simple. (As a veryspecial case, any differentiablysimple ring A with minimalideal is d-simplefor some derivationd; when A = Bn(F) this gives a result of Albert [3]). In ?6 the requiredfield E 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]for S[,,] to be D-simple. In ?8 we shall considerdifferentiably semisimple rings and give an ana- logue of the followingWedderburn-Artin theorem: a semisimpleassociative ringwith DCC on left ideals is a directsum of simple rings. 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 direct sum of simple rings then A/IL)is a direct sum of rings which are D-simple (or more preciselysimple 436 RICHARD E. BLOCK
with respectto the set of (quasi)-derivationsinduced on them by D). If A is associative and R is the (Jacobson)radical of A, then we say that A is D- semisimpleif 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 simi- lar 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 semisimpleflexible strictly power-associativealgebras are direct sums of simple algebras with a unit. The author has announcedelsewhere [41 the determination,by means of the Main Theorem,of the symmetrizedalgebra A+ forthe simpleflexible algebras in the finite-dimensionalcase. At the end of ?8 we shall state the generali- zation of this resultto the case of simple flexiblerings for which A+ has a minimalideal. For (finite-dimensional)Lie algebras of primecharacteristic it is nottrue that D-semisimplealgebras are direct sums of D-simple algebras since it is not even true for semisimplealgebras. Howeverin a semisim-pleLie algebra L everyminimal ideal I is (adL)-simple. This is the basis forthe application in ?9 of the Main Theoremto obtain a structuretheorem for semisimpleLie algebras. The author hopes that ideas connectedwith the Main Theorem will also proveuseful in the determinationof the simpleLie algebras.
2. Chains of ideals If A is an algebra over a ring K, we denote by T = T(A) the multipli- cation 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 {r,, lxi x e A} where rx and l1 are the right and left multiplicationsy v-iyx, y s--xy, respectively,of A (we do not assume that 1A e T (1A the identity 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 associative with 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-simplethen the D-centroidis a field;the proof is 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-simple if and onlyif A is D-simpleas a ring, again by the usual prooffor the case of ordi- narysimplicity. However fora suitable K thereexist K-algebras whichare differentiablysimple as a ring but not as a K-algebra, as we shall see in ?7 (we requirethat derivationsof a K-algebra be 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 both a 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 ele- ment d of HomK(A,A) such that [d, T] c T. Thus the set qderA of quasi- derivationsof A formsthe Lie normalizerof T in HomK(A,A). A quasi- derivationfor A as an algebra is also a quasi-derivationfor A 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 deri- vations. ( 1 ) If A is associative and I is an ideal, then (rxI I) e qderI for all x in A. ( 2 ) If c e C(A) and d e derA, then d + c e qderA and dc e qderA (while cd e derA). The above definitionsand facts (about the D-centroidetc.) go over forquasi-derivations as well. LEMMA 2.1. Let H be an associative algebra over a 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 some q > 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 the mapping m + Mj_1I- dm + Mq (m e Mj) is an isomor- phism a of Mj/Mj-1onto Mq+l/Mq. In particular Mq+,/Mq Ml. PROOF. If N is a submoduleof M then the mappingn v- dn + N of N into M/N is a homomorphismsince dhn + N = hdn + [d, hM]n+ N(h C H). 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 M1 is minimal and Mj/Mj-1_ Ml. This gives the requiredisomorphism of Mj/Mj-1onto Mq+i/Mq,and the lemma's proofis complete. Our firstapplications of this lemmawill be with M = A (an algebra over K), D a set of quasi-derivationsof A, and H= T(A), so that submodulesare the same as ideals. By a compositionseries of a ringor algebra,say of A, we shall mean (unless otherwisestated) a compositionseries for A as a T(A)- module,so that the termsof the series are ideals of A. 438 RICHARD E. BLOCK
LEMMA 2.2. Suppose that A is a quasi-differentiablysimple algebra 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 everycomposition factor is isomorphicto M1as a T(A)-module.
PROOF. Supposethat A does not have a compositionseries. Let d,. d... be a (possiblyempty) sequence of not necessarilydistinct quasi-derivations of A. Let i1 be the firstindex (if any) such that di1M,X M1 and use di1to define an ideal M2as in Lemma2.1. If M2,* * *, Mr have already been constructedby
Lemma2.1 using di1, * * *, dir-1 wherei1 : ... < ir-1 and dir-i . . . d2d1Mj c Mr let s = ir be the lowest index (if any) after it such that d8Mr .7 Mr. Then Lemma2.1 gives Mr,,. If dMr X Mr,,continue using d, until Mr-, .., * Mr+, have been obtained such that dsMr C Mr+fu Then ir+fuf_= s and d. * d2d1M1c Mr+u. Proceedinguntil the indiceshave been exhausted,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. ApplyingLemma 2.1 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. But the subspace of A spanned by all do . ** d1m(dne qder A, m e M1,n ? O0 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. Let N= M1_. Then N is maximal. Since A2 is a quasi-differentialideal (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. There- fore every productof at least 21-1elements of 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 centroid and 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 Jacobiidentity for commutators.It followsthat if d e qderA thenthe mapping co-o[d, c](c e C(A)) is a derivationof C(A). We shall denotethis derivationby d*. LEMMA 3.1. Let A be a quasi-differentiablysimple algebra over K. If A has a minimal ideal M1 and if 0 = M ,CiM1C * CiMl1 = Nc( M = A 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 followingholds for 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-, induced by 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 de- notesthe naturalprojection of A onto A/Miand pi is the injectionof Mi into A. The factors of r1(y)are T-homomorphismsand 1(y): A > A, so that a, maps F into C(A). If 0 # Y E r then y is onto and o1(7)A = M1,and hence a, is one-one. If 0 is a T-isomorphismof A/N onto M1 then p-10c F, (p-10) - p1Ozcl-,,and p-10is the requiredA. In the construction of M2, * **, Ml by Lemma 2.1, let di denote the quasi- derivationused to go fromMi to Mi+1,with ji the correspondingindex and 6t the correspondingT-isomorphism of Mj,/Mj,_1onto Mi+1/Mi. We define 2, * a,( recursivelyby setting oi,1() -[di, ri,(7)]. Thus ui, maps F into 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)N C Mi. If 0 is a T- isomorphismof A/N onto Mi?l/Mi,then bi10is a T-isomorphismof A/N onto 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 into Mj and ij- is 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 over K with a minimal ideal M1,where D c qderA. Then C(A) is D*-simple,where D* denotes{d* I d E D} (and d*(c) = [d, c] (c e C(A))). MoreoverC(A) has a compositionseries, with the same lengthI as that of A. PROOF. ( i ) If c e C = C(A) and cA c Mi (i > 0) (we use the notation of Lemma 3.1) thencN c Mi-1: Assumingthat cA 4 Mi-1,7ri-1c is a T-homo- morphismof A into A/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 cA c Mi, cA 0 Mi-, then thereexists a S in r such that (c - a(,8))A c Mi-1. Since cN c Mi-1, c induces a T-isomorphism0 of A/N onto Mi/Mi-1. By 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 I cA c M1} is a minimal ideal of C: It is obviously an 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 Lemma 3.1) whichinduces the T-automorphismO-'O' of A/N. (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 con- structionof the cr in Lemma 3.1 show that C -E ui(F) c H, and H = C. (v) The ideals E.=>,(r) -{ e C I cA c Mj0, .*, ,form a com- positionseries of C: The equality follows fromthe proofof (ii). The ex- pressionon the right side shows that theyare ideals, C being commutative. The expressionon the left side and the constructionof the ai in the proof of Lemma 3.1 show that the ideals are obtained by the methodof Lemma 2.1 startingfrom the 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 splits the exact sequence 0 N- A A/N -0 regarded as a sequence of E-module homomorphisms,then the mapping sO co-o c(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 over E. If r can be chosenpreserving products (that is, if the above sequencesplits when regardedas consisting of algebra homomor- phisms) thena' is a (C, E and K)-algebra isomorphism. PROOF. Since C/(radicalC) is a field,E acts unitallyon A and A is an E- algebra. 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 linearlyindependent over E, since if c = c; + E
that z-sj# 0. Since z-sjX N and cj induces a T-isomorphismof A/N onto MJ/M>_1,cjrsj i Mj-1. But A
4. The commutativeassociative case Suppose that A is a differentiablysimple commutativeassociative ring. At characteristic0, resultsof Posner [11] show that A is an integraldomain. In particularif A has a minimalideal then A is a field. For the application to the proofof the Main Theoremit wouldsuffice to provethis assuming that the radical N of A is nilpotent,A/N is a field,and A has a unit element;the proof in this case is trivial: If y C N, yi # 0, yi+sl 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-dimensionalalgebra over a fieldE and if A has the formA = El + N (N the radical),then A B-(E) forsome n > 0. The followingresult gener- alizes Harper's theoremin several ways. A portionof the proof is partly based on Harper's proof,but yieldsa shorterproof even of his special case. For the rings considered,the differentialconstants (i.e. elementsannihilated by all derivations)form a subfieldwhich may be identifiedwith the differential centroidunder a restrictionof the mappingx +- lx which identifiesthe ring withits centroid. THEOREM4.1. Let A be a differentiablysimple commutativeassociative ring of prime characteristicp, and let R ={x C A I xP = O}. If Rx = 0 for 442 RICHARD E. BLOCK 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 that A BJ(E) (- EJ), in fact iso- morphicas algebras over E. Here E may be taken to be any maximal sub- field 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 maximalsubfield of A containingF; let E be any such maximal 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 root of a, then xp- a is irreducibleover E and E(a) is a field,which contradictsthe maxi- mality 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 set of commutingindeterminates over E, and let z be the (unique) homomor- 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 { ii *. m+ Rkl--r|jrliX J, iI > 0, i+ *+ ijf k}
spans Rk/Rk-l-. This is true for k 1 by the choice of {yj}, and its truth for k + 1 followsfrom that for k because
RkR A5 (EyJi ... myj + Rk~l)(Eyi + Rdf) 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 obtained by 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 straightforwardverification shows, and this is the requiredd'. 'Added April 25, 1969. The writer has just learned of the paper by Shuen Yuan, Dif- ferentiablysimple rings of prime characteristic,Duke Math. J. 31 (1964).623-630, in which essentiallythe result of Theorem 4.1 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 Theorem self-contained. DIFFERENTIABLY SIMPLE RINGS 443
(iv) Let D' ={d'ederFE[XI }id ederA zd' = dz}; then ker z is a AmaximalD'-ideal and a derFE[X]-ideal of E[X]: kerz is obviouslya D'-ideal and is maximalby (iii) since A is differentiablysimple. Let I be the additive subgroupof E[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 ker I, or else I = E[XI, in which case kerz wouldcontain an elementwith a non-zero(monomial) term of 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) denotesthe ideal generatedby {XP Ij E J1 and the index set J is finite: (XP) c kerT. But (XI) is a maximal A-ideal of E[X] (just apply enoughderivations a/Dxj to an elementnot in (XP) to get a unit modulo(XP)). Hence kerz = (XP), and the nilpotencyof R impliesthat J is finite. This completesthe proofof the theorem. COROLLARY4.2. If a differentiablysimple commutativeassociative ring 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 J n. Then,as in (ii), RP' - RP"t, and RP=(R=)2 is a differentialideal. Hence R is nilpotent,and the resultfollows from the theorem. 5. Proof of d-simplicity If A is any algebra overa ringK, the quasi-derivationsform a Lie algebra over K undercommutation (the Lie normalizerof T(A)), as do the derivations. If c e C(A) and d e qderA then cd 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 ed C derA (and dc is a quasi-derivationbut notnecessarily a derivation). Thus 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 left multiplicationby elementsof 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)-compositionseries, M containsa minimalideal A1 of A. Starting with 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 algebra or 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 over K with a minimal ideal where D is a regular Lie subring of qderA thenA is d-simplefor some d 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 qder KA),we may ignoreK and regard A as a ring (the minimalideal remainsminimal). Also we mayassume that A is not simple. By Lemmas 5.1 and 3.2 we may assume that A is commutative associative and that D c derA. Then by Lemma 2.2 and Theorem4.1, A has primecharacteristic p 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 derivation d, 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 hence dir is a unit of A. Replacingd1 by (dlr)-'d, we mayassume that dlr I.1 Let Do- {d - (dr)dl1 d E D}. Then Do {d E D I dr = O} and 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 invariantunder d - (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-submodule invariant under di. Since H-submodulesare in particularideals of A, A has an H-composition series. Startingwith a minimalH-submodule M1 of A and applying Lemma 2.1 using d, at each step, we get an H-compositionseries 0 = MoC ** * c Ml = A and H-isomorphisms ti: Mi - Mi, where we write AMj= Mj/Mj-, (with 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 SIMPLE RINGS 445 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-simplethen A Bq(E) for somepositive q: By Theo- rem4.1, A Bq(E) forsome q > 0 with the same fieldE as above (this lat- ter fact is not actually needed) since A/N A/N. Since d1 gives all the mappingsQ,, if 0 # a e A then forsome i, dia X M11. But if q 0 then dia is a unit since A is a fieldand M,1 is nil, and hence any non-zerod1-ideal would containa unit. Now suppose q > 0, let x1,*--, xq be a set of nilpotentgenerators of A (i.e., A = E[x1, * Xi = O. i _ 1, *--, q; E a copy of E), and for i= 1, ..., I1 let Mi, M,' denote the ideals of A such that M, D M' D M _'D - '/AM,_= (elk **,)'l(x1, 6 * * *, Xq), and M1'/M1,._ (61o1... * )-'E(x, ... xq)p i.e., M'/Mi-,and MI'/Mi, correspondunder the H-isomorphismto the unique 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. Hence (d1w+ + Ml + M', and, by the Ml1-)(M' Ml1-) c M,1, (dlw)M' c H-isomorphisms,(d1w)M' 5 M''(i 1, ..., I1). Also wMi + Mi1 = M"' for all i sincethis holdsfor i 1. By the definitionof M' and M'', d1M' + Mi = M'.1 and d1M'' + M- M!'1 (i = 1, *--, I-1). Also wM' 0 since wM' c Suppose wM' c M'1 forsome i (1 < i < 1; this is truefor i = Ml-,. 1). Then (since [d1,I<] = ldjw) 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 AOIfor the ideal dI + I and doI + I, respectively, and similarlyfor a0 on ideals of A. Since A is do-simpleand has the same compositionlength as 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= M, (j = 1, , pq-1; i = 1, ,I). But for i < I, AM=-wd1M + M,= - wM,1 + Mll Ml' 1. Therefore the ideals jM7, j 1, ..., (pq-1)1, to- 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 differentiablysimple algebra over K with a minimalideal M, # A. Since derA is a regularLie ring,by Theorem5.2 thereis a deriva- 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 I d*c e I} is a subfieldof C with E + R -C. The differentialconstants of C are in E and in particularK4 c E. By ?4, C (and hence also A) has prime characteristicand 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-dimensionalover E. The ring A is also an algebra over E. In the notationof Lemma 3.1, if 0 vy P -C(A/N) then a1(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 isomorphismsa 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 isomor- phismof P onto I as E-modules. ThereforeA/N as an E-algebra is central. The subalgebra S' (la e A I da C M1} of Lemma 3.4 is closed underE, so we have the situationof Lemma 3.3, and A S' 0 EBfl(E) S't1]as an algebra over E, hence also over K, where S' A/N is a simple algebra over E and hence also over K. This completesthe proofof the Main Theorem.2 One of the main tools used in the proof of the Main Theorem was the passage fromquasi-derivations of A to derivationsof C(A). The proof of Lemma 3.4 was the only place where we required,for the D-simplealgebra A itself,that D consistof derivationsrather than merelyquasi-derivations. The followinggives several cases wherethis hypothesismay be weakened. COROLLARY6.1. Let A be a D-simple algebra with a minimal ideal, whereD c qderA. Then the conclusionof the Main Theoremholds in each of thefollowing cases ( i ) A has characteristic0; (ii) A is commutativeassociative with a unit element; (iii) A is finite-dimensionalover a field and is either alternative (in- cluding associative), or standard [1], [14] (including Jordan) of character-
2AddedApril 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 appropriate definitionsof 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) followsfrom Lemma 3.2 and the (easy) characteristic0 case of ? 4. Case (ii) follows fromLemma 3.2 and the Main Theorem since here A- C(A). Case (iii) follows fromthe Wedderburnprincipal theorem for alternative algebras [13] and for standard algebras [14] of character- istic # 2, since this providesthe splittingsubalgebra needed in Theorem3.3, thus bypassingLemma 3.4 (and ? 5). In Case (iv) the set S' of the proofof Lemma 3.4 is a subalgebra,and so as notedabove the conclusionholds in this case also. This concludesthe proof. The Main Theorem probablyremains true for any quasi-differentiably simplealgebra with a minimalideal. At any rate, Lemma 3.3 and Theorem 4.1, togetherwith Theorem5.2 and the argumentof ?6, providea great deal of informationabout such an algebra. It can also be seen that in the use we have made so far of the concept,the definitionof quasi-derivationcould be furtherweakened. Given one of the non-simplealgebras A of the Main Theorem,the algebra S is uniquelydetermined up to isomorphism,since it is the unique simpledif- ferencealgebra of A, and n is uniquelydetermined since pf is the composition lengthof A(p the characteristicof S and of A). One of the principaldifficul- ties 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 automorphismof A sendingone to the other. However the subalgebra is not in generalunique. To show this we need onlygive an automorphismof 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 deri- vation 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 Zassenhaus mentionedin ? 1. For any Lie algebra S overa fieldF of characteristicp and any integerm > 0 he con- structed[17, pp. 64, 67] a Lie algebra S(m',which he called a powerring of S. This consists of the vector space over F of all mn-tuplesa = (a,, , a.-,) of elementsof S, with productsgiven by [a, b] = c where Gt = .j0(')[ajj bi-j] (i = 0,1, *--. m-1) Based on his characterizationof the d-simpleLie algebras overF witha chief (= T(A)-composition)series [17, pp. 61-79],Zassenhaus conjectured [17, p. 80] 448 RICHARD E. BLOCK
that if L is a differentiablysimple Lie algebra over F (presumablywith a chief series) then L -Slv" for some simple Lie algebra S over F and some n > 0. It turnsout that we can show rathereasily (we omitthe details) that S(P%)and S[f,]= S X FBfl(F) are isomorphic,under the followingmapping
> (SS *s, p) - pT-i!-si' ? x{l ... an where i = i, + i2p + + ip"-', 0 < ij < p (i = 0, *.,p -1), denotes the residue class modulo p of l/pordPl, and Bn(F) = F[x1, *--, xJ, xn = 0 (j = 1, *- , n). Zassenhaus' conjecture then follows immediately from the Main Theorem. It seems clear that the tensorproduct form of Sn] used in the presentpaper is a muchmore convenientconstruction than 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 course in terms of the derivationsof S. We begin by determiningthe derivationsof a large class of tensorproducts.
THEOREM 7.1. Let A = S ( FB, regarded as an algebra over F, where F is a field, S and B are algebras over F, and B is commutativeassociative with a unit element. If S or B is finite-dimensionaland if S2 = S or {sGSIsS = Ss =O} 0 O then der A = (derS) ( FB + 1 0 F(der B) (the sum beinga direct sum 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) PROOF. A straightforwardverification shows that each d' 0 b and X0d" is a derivationof A. Now let d be a given derivationof A. We give the proofin threeparts. ( i ) Proof of thetheorem when S is associative witha unit element. We identifythe center of 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 index set J' = J U J" with J flJ" 0. Definelinear transformationsd' on S (i GI) and do' on B (j G J') by writing
d(s 0 1) = (d's) b , d(1 b) = Hje sjs0 (df'b) (0 (0 J, (seS, beB) DIFFERENTIABLY SIMPLE RINGS 449
Then each d' is a derivationof S since
d'(s8s2)I 0 bi = (d(s,0 1))(S2 0 1) + (s,0 1)d(S20 1) = S1{(dXs1)s2+ s,(d's2)}0 bi 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, [s, sj] (d;'b)= LjeJ. [8, sj] 0 (d!b) forall s in S and b in B. For a given b in B and each j 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 + S 0 d;'b bib , =.(ds) bib + Ejs sj d;'b (seS, beB) 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 for finitelymany of the indicesi. Thereforeif S is finite-dimensional,then d' 0 except for finitelymany i, so that E d, (0 bie (derS) 0 B, while of course the same is true if I is finite. Simi- larly Eie sj A) d;! e F 0 (derB). Finally, the sum is direct since if E, d 0 bie FN0derB then E, (d' 0 bi)(s0 1) 0 for all s, and d- 0 for all i. This completesthe proofwhen S is associative with 1. (ii ) If U = P + T(S) thenthere is 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 the multiplicationby b on B), and F and C(A) are commutative. U is an (associative)algebra of lineartransformations on S since71, =- 1, and yr8 rrs. There is a unique linearmapping z of U0 B onto a space of linear trans- formationsof A with z(u 0&b) = 0 lb (U e U, b e B). Then z is one-onesince if (L. ui 0 b4)(s0 1) = 0 forall s then ui = 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 (J(r(0 B) = C(A). Indeed if c e C(A) and we write c(s 0&1) = Ad y(s) (0 bi (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 -$(S) & bib,and c = Ad 7i bi. Here 7i O except for finitelymany i, as in (i). Therefore z7(F0 B) = C(A) and T(UO(& B) = C(A) + T(A). 450 RICHARD E. BLOCK
(iii) Proof of the theorem when S is not associative or does not have a unit element. Since [d, T(A)] c T(A) and [d, C(A)] C(A), commutation by d gives a derivation of C(A) + T(A), and hence by (ii) also a derivation d of U 0) B. The center of U is F since F is commutative. Since U is associative with a unit element, if {ya Ij C J} is a basis of F and {bi I i C I} of B then by (i) there are derivations d' of U and d;' of B such that
ld(s(@b) [d, ls b] =Zd(l8 (& b) =zTA. GI d~ti(l8)0@ bib + z jl0 9 d;!b (seS, beB)
Then with b 1 and with I) definedby setting d(s 01) = d'(s) 0 = d'(i e b- (so that as in (i) each d$ is a derivation of S) we have
l = 7 T ldi Lirs) 9 L) i 1 (s C S)
Hence Odo~s) = d for all i, and (since '(l,) 7jl=, 1
- l0d(s(3b)-,?Eidi(s)(@bib-A jgjrjs(@djbI ? (s e S, b C B) and the same equalityholds with l replacedby r. Then do d - d 0 bi -- jC 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 drop the hypothesisthat S2 -- S or {s e S I sS = Ss =O} 0= we would have to take into account summands d"' where d"'S2 0 and d"'S {sS I sS = Ss = 0}. We now apply Theorem7.1 to the determinationof derSr,,] when S is a 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 regardedhere as acting on A (and of course K4 c: F). In particu- DIFFERENTIABLY SIMPLE RINGS 451 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 der S -derFS. Also derFB"(F) is the well known npn-dimensionalJacobson-Witt algebra We,over F. We have now provedthe followingresult. COROLLARY7.3. If S is a simple algebra (of prime characteristic)over a ring K and if n > 0 then der S[},]= (derS) (&FBfl(F) + F OF Wn (regarded as acting on S OFBfl(F)), whereF is the differentialcentroid of Sr,], r is thecentroid of S, and W,.is the Jacobson-Witt algebra over F. 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 that d" is the component of d in derB, and denote it by dB. A straightforwardcomputation shows that ZdBb - [d, Zb1 for all 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 DB- simple, whereDB -{dB I d e D}. In particular, S[%1is differentiablysimple, and in fact d-simplefor some E-linear derivationd. PROOF. In provingthe firstconclusion we 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 linearlyindependent over 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 then S 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" where d" is the derivationof B given by ... d"r= (D/Dx1)+ xPj(Da/x2) + ... + (xi X.-.. -lax"') (where B = E[x1, . j,1 x? = 0, (y(a/xi)xj = 6jy(i, j = 1, ..., n)). Then B is d"-simple[3], d is K-and E-linear, and A is d-simple(just to show that A is differentiablysimple, it sufficesto use {is 0&D/Dx I i = 1, * , nn}). This completesthe proof. In discussingthe converseof the Main Theoremit remainsto show that SE,,,](S simple)has a minimalideal. IdentifySn] with S PBn(ZP),where 452 RICHARD E. BLOCK B.(Z)= ZAx1,*.*, I x.], X4 0 (i **., n). Then it is easy to see that S ?9 (x1 xX)-1 is a minimalideal of S[,, and in fact is containedin every non-zeroideal of S[.]. The followingis an example of an algebra A over a ring K such that A is differentiablysimple as a ring but not as an algebra over K. Let S be a central simple algebra over Z,, A = S (0ZB(Z ) for some n > 0, K 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 a K-algebra, has S 0 (xl, *.*, x) as a properdifferential ideal. 8. D-semisimplerings In this sectionwe obtain a D-structuretheorem which gives an analogue of the part of the Wedderburn-Artintheorem which says that a semisimple artinianring is a directsum of simplerings. We state the results for rings; they can 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 then the 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 quasi- derivationsof A/I, and by a slight imprecisionin language we shall speak of D-ideals of A/I ratherthan DA/[-ideals.Similarly if D consistsof derivations of A then D induces a set of derivationson I, and if D consistsof quasi- derivationsand I is a direct summandof A then D inducesa set of quasi- derivationson I; in eithercase, if I is (D I I)-simple we shall also say that I is D-simple. If the ideal I is an intersectionf M,of ideals of A then it is easy to see that ID n(Mk)Df If A is associative and R is the (Jacobson)radical of A then we call RD the D-radical of A. For alternativerings (which of course includeall associative rings)the radicalR is again takento be the intersection of the regularmaximal left 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 for everyx 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 alternativerings as 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 primitive ideals P of A. If A is a finite-dimensionalLie algebra then the radical R is taken to be the (unique) largest solvable ideal, and in a context in whichfinite-dimensional power-associative algebras are being discussedthe radical R is taken to be the (unique) largest nil ideal (this agrees with the DIFFERENTIABLY SIMPLE RINGS 453 previousdefinition in the alternativecase). In all these cases we call RD the D-radical of A and we say that A is D-semisimpleif RD 0O. 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 [D, HM]J 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 dXJIqt Mq, thenthere is a submoduleMq,+, with Mq D Mq+j, and an index j, 1 _ j _ q, such that dMq c Mj_1(Mo = M), dMq+lc Mj, and the mapping 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 re- strictionto Mq of the mappingm E-+dm + Mj is a homomorphismwith image Mj-1/Mjand kernelMq,+, and thus gives the requiredisomorphism. THEOREM8.2. Let I be an ideal of a ring A, let D be a set of quasi-deri- vations 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 e *.* e Sk (Si simple) then A/ID S1G1( ...* & SkGk where, for each i, Gi = 1 or Si has prime characteristicpi and Gi is a finiteele- mentaryabelian pi-group. PROOF. Withoutloss of generalitywe may assume that ID = 0. We first suppose that 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 cannot be ex- tended past some ideal Ml. Then Ml is a D-ideal and hence Ml = 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)-com- position series, any D-ideal L # A is contained in a maximal ideal I'. If I' zL I then I + I' = A and A/I'-I/I= n I' is nilpotent,contradicting A2= A. ThereforeI' = I D L, L = 0, and A is D-simple. Next suppose that A/I = (L1/I) E ... e* (Lk/I) wherethe Lj are ideals of A containingI such that Lj/I = Sj is simple(k is necessarilyfinite by DCc). For j = 1, ...* k, let P3 = Eioj Li. Then A/Pj (A/j)/(Pj/I) _ Sj is sim- ple, and A/PjD is D-simple with unique maximal ideal Pj. Moreover PlDfn fnPkD= (P1 n nPk)Dfl = ID. We may choose an index l < k and a reordering of the Pj such that PiD n ... n PID = ID and such that if 1 < i < 1 then Mj = n i=1, ,l;ioj PiD # ID. Considerthe homomorphismz of 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) alternative ring with DCC on ideals and with radical R, and let D be a set of quasi-derivationson A. If A is D-semisimple and 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 as- sociative or Cayleyrings [7]. Since AIR is artinianit is a finitedirect sum of simplerings by the same argumentas in the associativecase. Hence Theorem 8.2 gives the result. We now discuss a similarresult for an importantclass of power-associ- ative rings,the flexiblerings. A ring A is called flexibleif (xy)x = x(yx) for all x, y in A. Oehmke[10] provedthat a finite-dimensionalsemisimple flexible strictlypower-associative algebra 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,extending it to includethe characteristic 3 case as well as generalizingit to a resulton D-semisimplealgebras. In giv- ing this resultwe make use of the followingdefinitions and facts. Let A be an algebra over a fieldof characteristic# 2. Then A+ denotesthe algebra withthe same underlyingvector space as A and withnew multiplicationx *y = (1/2)(xy+ yx). For each x in A let dxbe the mappingof A into A definedby dxy= [x, y] (y e A). A directverification shows that each dx is a derivation 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}-ideal of A+. COROLLARY8.4. Let A be a finite-dimensionalpower-associative algebra over a field of characteristic# 2, with nilradical R and with a given (pos- sibly empty)set D of quasi-derivations,and supposethat A is D-semisimple. If A/R is flexibleand strictlypower-associative then A is direct sum of D- simple algebras (and is flexible,strictly power-associative and has a unit element). If A has a traceform then again A is a direct sum of D-simple algebras (and at characteristic# 5 is a non-commutativeJordan algebra). DIFFERENTIABLY SIMPLE RINGS 455 PROOF. The trace formcase follows immediatlyfrom Theorem 8.2 and Albert's theoremon trace forms[13, p. 136]. Also when AIR is commutative the result holds by Theorem 8.2 and the fact [2, 8] that semisimplecom- mutative strictlypower-associative algebras are directsums of simplealge- bras with a unit element(if S has unit elementthen so does SG) (the strict- ness of the power-associativityis relevant only 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 followsquickly from the flexiblelaw, as in [4], or, using power-associativity,from an idempotentde- composition.We may now apply Theorem8.2 to get the desiredresult. This completesthe proof. The decompositionof a D-semisimplering into D-simpleideals can also be provedeasily in the finite-dimensionalassociative case by using the mini- mal D-ideals. Before discoveringthe present version of Theorem 8.2 the authorhad used the minimalrather than maximal ideals to give a proofof Theorem 8.2 in the case when A is a finite-dimensionalpower-associative algebra and the Si have a unit element. This also gave Corollaries8.3 and 8.4. This was announcedbriefly in [5]. After[5] was submitted,the author received fromT. S. Ravisankar,of Madras, India, a copy of a recent manu- script which gives a proof,different from the author's, that if a finite-di- mensionalflexible strictly power-associative algebra of characteristic# 2, 3 is D-semisimple(D a set of derivations)then it satisfiesthe conclusionof the firstpart of Corollary8.4. He also establishes in another proof the trace formcase (again excludingcharacteristic 3). While discussingsemisimple flexible algebras we also mentionthe deeper questionof determiningthe structureof A+ for a simple flexiblealgebra A (of characteristic# 2). This was answered by the author [4] in the finite- dimensionalcase as an applicationof the Main Theoremof the presentpaper. Aside fromgiving a uniformproof of the various special cases proved by a numberof authors (see [4]), this solved the previouslyunsettled degree 2 characteristicp case, includedas well considerationof nil simplealgebras, and showedthat the resultdoes not dependon power-associativity.Since the Main Theorem has now been proved in a more general formthan when [41 was written,we now have the followingform of the resulton simpleflexible rings. 456 RICHARD E. BLOCK THEOREM8.5. Let A be a simple flexiblering of characteristic# 2. If A is not anti-commutativeand if A+ has a minimal ideal, theneither A+ is simple or the characteristic is prime and A+-B- (E) for somen > 0 and somefield E containingthe centroidof A. The proofis essentiallythe same as forthe finite-dimensionalcase in [4]. Since A+ is differentiablysimple, the Main Theoremimplies that if A+ is not simplethen the characteristicis primeand thereis a commutativesimple ring S such that A+ - S[Ef]for some n > 0. The problemthen is to show that S must be associative, and this may be done by showing that 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 finitedimensional semisimple Lie 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 pre- liminaryresult on the relation between L and its minimalideals. All the Lie algebras consideredare assumed to be finite-dimensionalover a field. For any Lie algebra L we writeinder L for the Lie algebra of inner derivations of L, i.e., inderL = {ad x I x E L}. LEMMA 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 many minimal D-ideals, say L1, ** *, Li, their sum M is direct, each Li is (D U inderL)-simple, the mapping x v--adMx(x E 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 generality we may assume that D is a sub- algebra 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 exactly the 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 maxi- mal directsum of minimalideals of L. Then M containsevery minimal ideal of L, the annihilatorin L of M contains no minimalideal of L (since any such would be abelian) and so is 0, and hence x e->ad~x is an isomorphismof L into derM. The remainingconclusions follow easily. This completesthe proof. DIFFERENTIABLY SIMPLE RINGS 457 In the case in whichD - derL, the resultsof Lemma 9.1, exceptfor the 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 differentiablysimple and so any ideal of L properlycontained in 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 differentiablysimple algebras and their deri- vationalgebras leads quicklyvia Lemm 9.1 to the followingdescription of 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 S = > Si 0 Bi whereBi denotesBn.(F) and Bi = F if ni = 0 (all algebras and tensorproducts considered here are overF), and identifyS with inderS, so that S = inderS = (inderSi) (0 Bi c der S er1 ((der Si) 0 Bi + Pi 0 derBi) whereFi denotesthe centroidof Si. Now let L be any subalgebra of derS containing S (hence L is uniquely determined by a subalgebra of e3r=1 ((outder Si) 0 Bi + Fi 0 der Bj)). For i 1, ** , r let Li denotethe set of componentsin der(Si 0 Bi) of elementsof L, and if S is central,so that der(Si 0 Bi) = (der Si) 0 Bi + 1,. 0 der Bi, let LB. denote the set of com- ponentsin derBi of elementsof Li (in the terminologyof ?7). THEOREM 9.3. Every finite-dimensional semisimple Lie algebra of prime characteristicis isomorphicto one of the algebras L just constructed, and the semisimple algebra uniquely determinesr and the pairs (Si, ni), i 1, . *, r, up to isomorphismand reordering. The algebra L constructed is semisimpleif and only if Si 0&Bi is Li-simple (which when Si is central is equivalent to Bi being LBi-simple) for i = 1, *- -, r. The mapping x e->ad1lx (x e N,,, L) is an isomorphismof the normalizerof L in derS onto derL. The same results hold with differentiablysemisimple in place of semisimple provided the condition on Li (or LB.) is replaced by the same condition on (Ndr sL)i(or(NdersL)Bi)- PROOF. By Lemma 9.1 any semisimplealgebra withsocle Mis isomorphic 458 RICHARD E. BLOCK to a subalgebraof derM containinginder M, and by the Main TheoremM is isomorphicto some S of the above formfor suitable pairs (Si, ni), thesebeing essentiallyuniquely determinedby 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 nil- potent. This proves the second sentence. The mappingz definedby z-x adLx(xe N. LL) is a homomorphisminto derL. If adLx 0 then ad~x = 0 and x = 0. Hence z is one-one. Suppose d E derL and let x be the unique element of der S 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 finalstatement may be proved the same as the firsttwo by replacingsemisimple, socle, ideal by differenti- ably semisimple,etc., and L-simple by (der L)-simple,using the characteri- zation of derL just obtained. This completesthe proof. COROLLARY9.4. Let F be a field of characteristic p. If every simple Lie algebra over F of dimension < m has all its derivations inner then everysemnisimple Lie algebra over F of dimension < min{m + 1, 3p} is a direct sum of simple algebras. This is an immediateconsequence of Theorem9.3, and gives a generali- zationof 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 that startingfrom a 3-dimensionalcentral simple Lie algebra over F, we can constructa semisimpleLie algebra of dimension3p + 1 and a perfectsemi- simpleLie algebra of dimension4p, neitherof whichis a directsum of simple algebras. UNIVERSITY OF CALIFORNIA, RIVERSIDE REFERENCES [ 1I A. A. ALBERT, Power-associativerings, Trans. Amer. Math. Soc. 64 (1948),552-593. [2] , A theoryof power-associativecommutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503-527. [3] , On commutativepower-associative algebras of degreetwo, Trans. Amer. Math. Soc. 74 (1953), 323-343. DIFFERENTIABLY SIMPLE RINGS 459 4] R. E. BLOCK, Determinationof A+ for the simple flexiblealgebras, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 394-397. [5] , Differentiablysimple algebras, Bull. Amer. Math. Soc. 74 (1968), 1086-1090. [6] L. R. HARPER, JR., On differentiablysimple algebras, Trans. Amer. Math. Soc. 100 (1961), 63-72. t 7 ] E. KLEINFELD, Primitivealternative rings and semisimplicity,Amer. J. Math. 77 (1955), 725-730. [8] L. A. KOKORIS, New results on power-associativealgebras, Trans. Amer. Math. Soc. 77 (1954), 363-373. [9] A. I. KOSTRIKIN, Squares of adjoint endomorphismsin simple Lie p-algebras,Izv. Ak. Nauk SSSR Ser. Mat. 31 (1967), 445-487(in Russian). [10] R. H. OEHMKE, On flexiblealgebras, Ann. of Math. (2) 68 (1958), 221-230. [11] E. C. POSNER, Differentiablysimple rings, Proc. Amer. Math. Soc. 11 (1960), 337-343. [12] R. REE, On generalized Witt algebras, Trans. Amer. Math. Soc. 83 (1956), 510-546. [13] R. D. SCHAFER, An Introduction to Nonassociative Algebras, New York, Academic Press, 1966. [14] , Standard Algebras,Pac. J. Math. 29 (1969), 203-223. [15] A. SAGLE and D. J. WINTER, On homogeneousspaces and reductivesubalgebras. of simple Lie algebras, Trans. Amer. Math. Soc. 128 (1967), 142-147. [16] G. B. SELIGMAN, Characteristicideals and the structureof Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 159-164. [17] H. ZASSENHAUS, Uber Lie'sche Ringe mit Primzahlcharakteristik,Abh. Math. Sem. Hamburg 13 (1939), 1-100. (Received September 17, 1968.)