Julius Zelmanowitz

The finite intersection property on annihilator right ideals

In this article the finite intersection property on annihilator right ideals will be shown to be an adequate substitute for more stringent chain conditions on such ideals. One application of the investigation will produce a new characterization of orders in semisimple artinian rings, another will generate new classes of absolutely torsion-free rings. Until otherwise indicated all rings are arbitrary associative rings not necessarily possessing an identity element. To simplify the statements of the results, we expand the usual definition of a prime ideal to include the ring itself. For any subset A of a ring R we set r(A) = (0: A) = {r E R lAr = O}, the right annihilator of A; (0: x) being written for (0: {x}). More generally, for A and B subsets of R, (B: A) will denote {r E RIAr C B}. We let 1(A) denote the left annihilator of A. A ring R is said to have the finite intersection property on right annihilators provided that whenever r(A) = 0 for a right ideal A C R there exists xl, . . ., x E A withn 2 1(?: xi) = 0. It is readily apparent that a ring which satisfies the descending chain condition on right annihilators possesses this property; for choosing xl, . . ., x,P e A with n n= 1(0: xi) minimal among all such intersections forces n n= I (0: xi) = 0. The converse is false however. For instance a commutative subdirectly irreducible (i.e., having a unique minimal ideal) nil ring which is not nilpotent has the finite intersection property; in fact it satisfies the stronger requirement that nxEA (0: x) = 0 implies that (0: x) = 0 for some x E A. But such a ring cannot satisfy the descending chain condition on annihilators, else by well-known theorem [3, Theorem 1] it would be nilpotent. For a specific example of such a ring one may take any subdirectly irreducible homomorphic image of EDO<,,Fx, where F is a field and multiplication is defined by xax,I = xa+,1 if a + 38 < 1 and 0 otherwise (see Example 3 of [1]). This example also demonstrates that finite intersection properties on annihilators cannot force the nilpotence of nil rings. A ring will be called nonsingular if its right singular ideal Z(R) is zero, where Z(R) = {a e RI(0: a) is an essential right ideal). PROPOSITION 1. Assume (1) r(P) = O for every prime ideal P of R; and (2) R has the finite intersection property on right annihilators. Received by the editors December 18, 1974 and, in revised form, March 27, 1975. AMS (MOS) subject classifications (1970). Primary 16A34; Secondary 16A12, 16A18, 16A46.

An extension of the Jacobson density theorem

The purpose of this note is to outline a generalization of the Jacobson density theorem and to introduce the associated class of rings. Throughout R will be an associative ring, not necessarily possessing an identity element. MR will always denote a right R-module, and homomorphisms will be written on the side opposite to the scalars. For an element m G M, set (0:m) = {rGR\mr = 0}. A nontrivial module MR is called compressible if it can be embedded in any of its nonzero submodules. A compressible module MR is critically compressible if it cannot be embedded in any proper factor module. For a compressible module MR, each of the following conditions is equivalent to MR being critically compressible: (1) MR is monoform (i.e. nonzero partial endomorphisms of M are monomorphisms); (2) MR is uniform and nonzero endomorphisms of M are monomorphisms. The proof of these characterizations is elementary. In the absence of more suitable terminology let us define a ring to be weakly primitive if it possesses a faithful critically compressible module. A weakly primitive ring is prime; for it is easy to see that the annihilator of a compressible module is a prime ideal. In order to simplify this presentation let us call (A, ± VR, MR) an Rlattice if F is a A-R bimodule where A is a division ring, AM = V, and R acts faithfully on M (so that R can be regarded as a subring of End A V). We are now prepared to state the main result.

Artinian rings with restricted primeness conditions

Abstract Complete blocked triangular matrix rings over division rings are characterized as left artinian rings which satisfy suitably restricted primeness conditions. A matrix representation is also provided for the class of nonsingular left artinian C-prime rings.

Semiprime modules with maximum conditions

A module RM is semiprime if for each 0 ≠ m ϵ M there exists ƒ ϵ HomR(M, R) with (mƒ)m ≠ 0. In Section 1 semiprime artinian modules are seen to be isomorphic to finite direct sums of minimal left ideals generated by idempotents. Semiprime noetherian modules have endomorphism rings which are left orders in semisimple artinian rings; and necessary and sufficient conditions for the latter situation to occur are given in Section 3. Prime modules are defined analogously and are treated simultaneously; and the above results are actually considered in the broader milieu of Morita contexts. In Sections 4 and 5 the classical density theorem for rings with faithful minimal left ideals is generalized (with a weakened definition of density) to include semiprime rings possessing faithful finite dimensional left ideals. The method of proof covers the infinite dimensional case as well. As a consequence, the classical density theorem is extended to rings with faithful completely reducible left ideals. In Section 6, the endomorphism ring of a torsionless module over a dense ring of transformations is shown to be a ring of the same type.

Representation of rings with faithful polyform modules

This article studies subrings which satisfy a density-type criterion called m-density. It is first observed that if V is a faithful quasiinjective R-module then R is an m-dense subring of BiendRV. This is used to obtain the main result which states that R is an m-dense subring of a direct product of rings of linear transformations if and only if R has a faithful locally finite dimensional module whose essential sub-modules are rational.

On matrix equivalence and matrix equations

Abstract Roths theorem on the solvability of matrix equations of the form AX−YB=C is proved for matrices with coefficients from a division ring or a ring which is module-finite over its center.

Semisimple rings of quotients

Necessary and sufficient conditions on an arbitrary Gabriel filter of left ideals of a ring R are determined in order that the ring of quotients of R with respect to the filter be semi-simple artinian. Special instances include generalizations of earlier work on classical rings of quotients and maximal rings of quotients.

Correspondences of closed submodules

If N is an M-faithful R-module, then there is an order-preserving correspondence between the closed R-submodules of N and the closed Ssubmodules of HomR(M, N), where S = EndRM. There is a considerable body of research dealing with correspondences between the lattice of submodules of an R-module M and the lattice of left ideals of its endomorphism ring S. This literature includes the well-known Morita theory and its generalizations. When a complete correspondence between all submodules and all left ideals fails to hold, one may still ask whether there is a correspondence between designated sublattices of submodules and left ideals. One particular situation that has attracted much attention has been cases when there is a correspondence between the closed (i.e., essentially closed) submodules of M and the closed left ideals of S. It is known that such a correspondence exists when M is a semisimple module (an elementary observation), when M is a free module [2], when M is a nonsingular retractable module satisfying an additional condition [7], and when M is a nondegenerate module [8]. This article began as a search for a conceptual link among these special cases. Eventually, it was realized that a common denominator is the notion of a self-faithful module, a concept first introduced for generators in [6], and recently exploited to good effect in [3], [4] and [5]. The principal contribution of this article is to demonstrate that a natural correspondence of closed submodules with closed left ideals occurs whenever M is a self-faithful module. In fact, taking a cue from the approach in [1], we show more generally in Theorem 1.2 that when N is an M-faithful R-module, then there exists an order-preserving correspondence between the closed R-submodules of M and the closed S-submodules of HomR(M, N), where S = EndRM. Taking N = M then specializes to the self-faithful case. Additional examples of M being self-faithful, and of the desired correspondence holding, occur when M is a quasi-projective retractable module (Proposition 1.2 in [3]) and when M is a polyform retractable module (Corollary 2.3). Of purely technical interest is the fact that the results contained in this paper remain true even over rings which fail to have an identity element. Received by the editors March 7, 1995. 1991 Mathematics Subject Classification. Primary 16S50; Secondary 16D70. ?)1996 American Mathematical Society

A Class of Modules with Semisimple Behavior

Weakly semisimple modules were introduced in [11] as a simultaneous generalization of semisimple modules and of monoform compressible modules. A principal goal of this paper is to show that the quasi-injective hull of a weakly semisimple module, when viewed in terms of its lattice of quasi-injective submodules, exhibits the behavior of a semisimple module.

Quotient rings of endomorphism rings of modules with zero singular submodule

Throughout this paper (R, M, N, S) will denote a Morita context satisfying a certain nonsingularity condition. For such contexts we give necessary and sufficient conditions in terms of M and R for S to have a semisimple maximal left quotient ring; respectively a full linear maximal left quotient ring, a semisimple classical left quotient ring. In doing so we extend the corresponding well-known theorems for rings (employing them in the process) to endomorphism rings. Suppose (R, M, N, S) is a Morita context ([1], [2]). That is suppose RMs and sNR are bimodules with an R-R bimodule homomorphism (, ): M Os N-*R and an S-S bimodule homomorphism [,]:N OR M-*S satisfying m[n1, n2] = (ml, n1)M2 and n1(ml, n2) = [n1, m1]n2 for all ml, m2 E N and nl, n2e N. Throughout, unless otherwise indicated, M and N will satisfy the following condition: Ms is faithful; and [N, m] =0 for m E M implies that m=0. Note that when this condition is satisfied, we can (and will) assume that ScI HomR(M, M). Let RM be any left R-module, and set N=HomR(M, R) and S= HomR(M, M). Set (i,f)=(m)f for m E M,fe N; and [f, m] is defined via m1[f, m]=(m1,f)m for all m, ml E M, f E N. Then (R, M, N, S) is a Morita context, called the standard context for RM. If R is semiprime and RM is torsionless, then the above condition is satisfied by the standard Morita context for RM. If RM is a generator and 1 E R; or indeed, if (Trace RM)MOO whenever Oom E M, then the standard Morita context for RM satisfies the above condition. LEMMA 1. (a) If A is an essential left ideal of S, then MA is an essential submodule of RM. Received by the editors November 8, 1971. AMS 1970 subject class{fications. Primary 16A08, 16A64; Secondary 16A12, 16A42. Key w1ords and phrases. Endomorphism rings, Morita contexts, singular submodule, quotient rings, full linear rings, Goldie dimension. 0 American Mathematical Society 1972