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Determination of the Differentiably Simple Rings with a Minimal Ideal Author(S): Richard E 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 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=annals. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics. 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).
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