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LetR be a ring, G a finite group of automorphisms acting on R, and RG the-fixed subring of R. We prove that if R is semiprime with no additive ¦ G¦-torsion, then R is left Goldie if and only if RG is left Goldie. By coupling this with an examination of the prime ideal structures of RG and R, we are able to prove that if ¦G ¦ is invertible in R and RG is left Noetherian, then R satisfies the-ascending chain condition on semiprime ideals, every semiprime factor ring of R is left Goldie, and nil subrings of R are nilpotent. For the pair RG and R, we also consider various other properties of prime and maximal ideals such as lying over, going up, going down, and incomparability.

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

In this paper we show that a ring R is von Neumann regular (or a V-ring) if and only if every central localization of R at a maximal ideal of its center is von Neumann regular (or a V-ring). Strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. Also we consider whether regular PI-rings can be characterized by the property that all central localizations at maximal ideals of the center are simple. Commutative von Neumann regular rings have been characterized in various ways. However, very few of these characterizations extend to noncommutative rings. The results in this paper arose from attempting to extend to noncommutative rings a well-known theorem of Kaplansky [11, Theorem 6] which states that commutative regular rings are characterized by the property that all localizations at maximal ideals are fields. It turns out that the obvious extension of this theorem to the noncommutative case is valid. That being, a ring is regular if and only. if all central localizations at maximal ideals of the center are regular. An analogous theorem is obtained for Vrings. Also we show that strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. With an eye to the commutative theory, we consider whether regular PIrings can be characterized by the property that all central localizations at maximal ideals of the center are simple. We provide an example to show that this is not the case. However, it is true if and only if contraction provides a 1:1 correspondence between maximal ideals of the ring and maximal ideals

A classical theorem of E. Noether asserts that if R is a commutative ring, finitely generated over a field k, and G is any finite group of k- automorphisms of R, then the fixed ring (or ring of invariants) R” is also finitely generated. The question naturally arises as to what extent Noether’s theorem can be generalized to the noncommutative case. If R is also Noetherian and IG/ ~ ’ E k, all is well: RG is finitely generated, a result of Montgomery and Smail [6]. However, it is false in general, even for Pf rings [6]. Moreover, a recent result in Dicks and Formanek [l] (and, somewhat later, Kharchenko [4]), shows that almost the opposite of Noether’s theorem holds in the free aIgebra. That is, they prove that if G acts linearly on the free algebra F= k(xj,..., x,>, then P is finitely generated if and only if G acts by scalar matrices. In the present paper we consider the analogous problem for an algebra of generic matrices. That is, let U = k(X i,..., X,f be the generic matrix algebra generated over a field k by the m x m (m 2 2) generic matrices X1,..., X, (da 2). Let G act linearly on U; that is, for each gE G, Xp = c, cliiXi, for uij E k. Thus g corresponds to the dxd matrix A = (a,). If G consists of scalar matrices and \G/ -’ E k, then UC is always finitely generated. For, consider the free algebra F= k(x, ,..., xd> with the same action; since I/= ir’, a homomorphic image of F, it follows that v =

Kaplansky posed the following question: Are prime von Neumann regular rings primitive? We show that with a certain countability condition the answer is affirmative. This is used to simplify and clarify earlier work on the existence of primitive group algebras. In [7] Kaplansky posed tfie following question: Are prime von Neumann regular rings primitive? Our main theorem in ?1 asserts that a prime von Neumann regular ring with a countable cofinal subset of ideals is indeed primitive. From this it follows that Kaplansky's question has an affirmative answer for each of the following classes of rings: (1) countable rings, and (2) rings with the descending chain condition on two-sided ideals. We call locally Artinian a ring which is the union of an ascending sequence of left Artinian subrings. In ?2 we establish that a semisimple locally Artinian ring is primitive if and only if it is prime. This elucidates and extends earlier work of Formanek and Snider [5] and Passman [11] on the existence of primitive group algebras. Moreover, we apply it to produce new primitive group algebras by showing that the group algebra over any field of the group of permutations on an infinite set which move only finitely many elements is primitive. 1. Prime von Neumann regular rings. Let R be a prime ring. The unadorned word "ideal" will always mean "two-sided ideal". Under the partial order of inclusion the set S of nonzero ideals of R is a directed set since I, J in S implies InrJ in S, I rJc I, and InJciiJ. We say that R has a countable cofinal subset of ideals if there exists a countable subset C of S so that for each A in S there is an I in C such that Ic A. We will be considering a ring R in which each nonzero right ideal contains a nonzero idempotent. Note that this is equivalent to each nonzero left Presented to the Society, April 28, 1973 under the title Prime regular rings; received by the editors February 16, 1973. AMS (MOS) subject classqifcations (1970). Primary 16A26, 16A30; Secondary 16A12, 16A20, 16A32.

(1978). On injective and surjective endomorphisms of finitely generated modules. Communications in Algebra: Vol. 6, No. 7, pp. 659-672.

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

Both the classical approach to decomposition theories and Fisher’s technique of constructing decomposition theories from radical functions are extended to and exploited in the context of abelian categories. These two different approaches to decomposition theories for abelian categories intertwine in one theorem from which flows necessary and sufficient conditions for the existence of the tertiary, primary, and Bourbaki’s P \mathcal {P} -primary decomposition theories.

Does Hopkins' theorem extend to projective modules, i.e., are projective Artinian modules Noetherian? An example is given to answer this question in the negative; however, we show that the answer is affirmative for certain large classes of projective modules. Dually, are injective Noetherian modules Artinian? Again the answer is negative; nevertheless, we provide an affirmative answer for certain classes of injective modules. Introduction. It is well known that the endomorphism ring of a module which is both Artinian and Noetherian is semiprimary. The author has noted [7] that the endomorphism rings of both projective Artinian and injective Noetherian modules are semiprimary. This raised the following questions: Do there exist projective Artinian modules which are not Noetherian, or could it be true that Hopkins' theorem [11, p. 132] extends to projective modules? Dually, do there exist injective Noetherian modules which are not Artinian [8], [9, p. 378]? In ?1 we prove that the endomorphism rings of both projective Artinian and injective Noetherian modules are semiprimary. We give an example of a projective Artinian non-Noetherian module in ?2 and show that Hopkins' theorem does extend to a projective Artinian R-module M in each one of the following cases: (a) R is commutative, (b) R is hereditary, or (c) M is a generator in the category of R-modules. An example of an injective Noetherian non-Artinian module is given in [14]. In ?3 we prove that if M is an injective Noetherian R-module where R is a ring with polynomial identity which satisfies the ascending chain condition on annihilators of submodules of M, then M contains an essential submodule which is Artinian. From this it follows that injective Noetherian modules over commutative rings are Artinian. Presented to the Society, March 3, 1971 under the title Endomorphism rings of modules; received by the editors September 29, 1972 and, in revised form, December 27, 1972. AMS (MOS) subject classifications (1970). Primary 16A50, 16A52, 16A46; Secondary 16A64, 16A38, 18E10.

For a ring R R which satisfies a polynomial identity we show that the following are equivalent: (1) R R is von Neumann regular, (2) each two-sided ideal of R R is idempotent, and (3) each simple left (right) R R -module is injective. We show that a P.I.-ring R R is left perfect if and only if all left R R -modules have maximal submodules and R R has no infinite sets of orthogonal idempotents.

Introduction. Lesieur and Croisot in [7] have generalized the classical primary decomposition theory for Noetherian modules over commutative rings to the tertiary decomposition theory for Noetherian modules over rings, which are not necessarily commutative, but which have a certain chain condition on ideals. Riley has shown in [8] that for finitely generated unitary modules over left Noetherian rings with identities, the tertiary decomposition theory in a certain sense is the only natural generalization of the classical primary theory. The purpose of this paper is to show that the tertiary decomposition theory extends to a larger family of modules. We call this family the family of t-worthy modules. In particular we show that for an arbitrary ring R, the tertiary theory holds for any R-module M which has the property that each factor module of M is finite dimensional in the sense of Goldie. For the family of t-worthy modules we show that, with certain reasonable assumptions, the tertiary theory is the only theory that provides all the salient features of an ideal theory in the sense of the classical primary theory. In order to make this extension, we introduce the abstract concepts of radical functions, associated ideal functions, and decomposition theories. From a radical function we construct a decomposition theory on a certain family of modules. Moreover we show that any decomposition theory can be constructed in this way from a radical function. Therefore our technique appears to be the natural way to construct decomposition theories. Hence we proceed by constructing decomposition theories in this way from various radical functions. In ?5 we construct a decomposition theory from the tertiary radical function, t, and prove that it is the tertiary decomposition theory. Moreover we show that a necessary and sufficient condition for the tertiary decomposition theory to exist on an admissible family C of modules is that each module in C be t-worthy. Goldie's dimension theory is then used to produce examples of admissible families of t-worthy modules. An example is given in ?6 to show that the associated ideals, which are used to produce the extension of the tertiary theory, need not be prime ideals. Also it is shown that in certain special cases, our definition of associated ideal specializes to the usual definitions of associated prime ideal.

On June 25, 1962, in Engel v. VitaleJ the Supreme Court of the United States declared the New York Regents' Prayer to be in violation of the establishment clause of the First Amend ment to the Constitution of the United States. That date was immediately designated "Black Monday," implying that the Court, in arbitrary fashion, had thrown to the winds the nation's hard won and greatly treasured religious freedom. Sober re flection and study shows that what was originally regarded as a decision against religious freedom was in reality a decision for religious freedom. The Regents' Prayer—"Almighty God, we acknowledge our dependence upon Thee, and we beg Thy blessings upon us, our parents, our teachers and our country"—was commissioned, adopted and promulgated for use in the public schools of New York by that state's highest educational authority, the State Board of Regents. Challenged by five families of varying re ligious faiths—Ethical Culture, Jewish, Unitarian—and of no religious faith, Engel v. Vitale went into the Reports of the United States Supreme Court thereafter to denote an excellent, if unconstitutional, example of government religion.2 The Supreme Court divided six to one in Engel v. Vitale. Justice Hugo Black, speaking for the Court, first demolished the argument that, although religious in nature, the Regents' Prayer was based on the "spiritual heritage" of the nation and was, therefore, without constitutional defect. By reviewing the history of the church-state relationship in England and the American colonies, Black demonstrated that the nation's spiri tual heritage, additionally if not primarily, demands separation of church and state for the benefit of both. The defenses of non denominational and non-compulsory were also unavailing. The

In this paper we prove that if G is a finite group of automorphisms acting on a semiprime ring R such that R has no additive ] G j-torsion, then the skew group ring R*G is also semiprime. The result was heretofore known in such special cases as when G is finite abelian, R is Goldie, or R satisfies a polynomial identity [I]. Our technique of proof is to show first that (even in the presence of 1 G I- torsion or G possibly infinite) every nonzero ideal in R*G contains a nonzero element whose support consists of inner automorphisms, where inner is used in the sense of Kharchenko [5]. Using this we are able to show that R*G is semiprime provided that R is prime and B*Ginn is semiprime where Glnn is the subgroup of inner automorphisms and B is the algebra of the group in the quotient ring of R with respect to the filter of two-sided ideals. Next we show that indeed B*Ginn is semiprime when R has no additive 1 G j-torsion. Finally, in order to complete the proof, we make a reduction from R semiprime to R prime. Along the way we prove that if G (possibly infinite) is outer and

In this paper we make an investigation into the structure of semiprime polynomial identity rings which is culminated by showing that each such ring R R has a unique maximal left quotient ring Q Q such that (1) Q Q is von Neumann regular with unity and (2) every regular element in R R is invertible in Q Q .

The connections between the prime ideal structure of a ring and that of its subring fixed by the action of a finite group of automorphisms is studied. Such a study was begun by Osterburg and the author in a paper with the same title. Recent work by Lorenz and Passman has added new impetus to the study. We will look at their work together with its ramifications.

In a paper with a similar title Herstein has considered the structure of prime rings which contain an element a which satisfies either[a,x]n=0or is in the center ofRfor eachxinR. We consider the structure of rings which contain an element a which satisfies powers of certain higher commutators. The two types which we consider are (1)[…[[a,x1],x2],…xm]n=0or is in the center ofRfor allx1,x2,…,xminRand (2)[a,[x1,[x2,…,[xm−1,xm]…]]]nfor allx1,x2,…,xminR. We obtain results similar to those obtained by Herstein; however, in some cases we must strengthen the hypotheses.Also we consider a third type (3)(axm−xna)k=0for allxinRand extend results of Herstein and Giambruno.

Let M be an R-module and let EndR(M) be the ring of all R-endomorphisms of M. If Mis Artinian, then each nil subring of EndR (M) is nilpotent. If M is Noetherian, then the indices of nilpotency of the nil subrings of EndR(M) are bounded. Introduction. In [6] Procesi and Small showed that if M is a Noetherian R-module and if R satisfies a polynomial identity, then each nil subring of EndR(M) is nilpotent. Later Small [4] removed the hypothesis on R and proved that if M is Noetherian, then each nil subring of EndR(M) is nilpotent. This raises the dual question: If M is Artinian, then is each nil subring of EndR(M) nilpotent? We give a proof which does provide an affirmative answer; moreover, it can be immediately dualized to give a proof of Small's theorem. ?2 is an appropriate place to include the following joint result of the author and Lance W. Small (who kindly gave his permission to me to publish it here): If M is a Noetherian R-module, then the indices of nilpotency of the nil subrings of EndR(M) are bounded. In proving this we also show that if M is an injective Noetherian R-module, then the Jacobson radical of EndR(M) is nilpotent. Putting this together with wellknown results, we obtain that the endomorphism ring of an injective Noetherian module is semiprimary. At the end of the paper the author poses two questions which arise from this work. 1. Endomorphism rings of Artinian modules. Throughout this paper, R will denote an associative ring which does not necessarily have a unity and M will denote a right R-module. LEMMA 1.1. If M is an Artinian R-module, then S=EndR(M) satisfies the ascending chain condition on left annihilators. PROOF. We note that Ml becomes a faithful left S-module via m= +(m), b E S, m E M. In order to show that S satisfies the ascending chain Received by the editors October 26, 1971. AMS 1970 subject classifications. Primary 16A22, 16A34; Secondary 16A64, 16A52.