S. Tariq Rizvi

Baer and quasi-baer modules

Abstract We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) 𝒦-cononsingular if and only if it is (FI-) extending and (FI-) 𝒦-nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.

Modules in Which Every Fully Invariant Submodule is Essential in a Direct Summand

Abstract A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring.

ON -NONSINGULAR MODULES AND APPLICATIONS

We introduce the notion of 𝒦-nonsingularity of a module and show that the class of 𝒦-nonsingular modules properly contains the classes of nonsingular modules and of polyform modules. A necessary and sufficient condition is provided to ensure that this property is preserved under direct sums. Connections of 𝒦-nonsingular modules to their endomorphism rings are investigated. Rings for which all modules are 𝒦-nonsingular are precisely determined. Applications include a type theory decomposition for 𝒦-nonsingular extending modules and internal characterizations for 𝒦-nonsingular continuous modules which are of type I, type II, and type III, respectively.

Dual Rickart Modules

Rickart property for modules has been studied recently. In this article, we introduce and study the notion of dual Rickart modules. A number of characterizations of dual Rickart modules are provided. It is shown that the class of rings R for which every right R-module is dual Rickart is precisely that of semisimple artinian rings, the class of rings R for which every finitely generated free R-module is dual Rickart is exactly that of von Neumann regular rings, while the class of rings R for which every injective R-module is dual Rickart is precisely that of right hereditary ones. We show that the endomorphism ring of a dual Rickart module is always left Rickart and obtain conditions for the converse to hold true. We prove that a dual Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is a dual Baer module. A structure theorem for a finitely generated dual Rickart module over a commutative noetherian ring is provided. It is shown that, while a direct summand of a dual Rickart module inherits the property, direct sums of dual Rickart modules do not. We introduce the notion of relative dual Rickart property and show that if M i is M j -projective for all i > j ∈ ℐ = {1, 2,…, n} then is a dual Rickart module if and only if M i is M j -d-Rickart for all i, j ∈ ℐ. Other instances of when a direct sum of dual Rickart modules is dual Rickart, are included. Examples which delineate the concepts and results are provided.

On injective and quasi-continuous modules

We reprove that any injective module has a direct decomposition into a directly finite and a purely infinite part ([8], Theorem 6), and we show that this decomposition has a strong uniqueness property (Theorem 1). As a consequence of these facts, and of the cancellation property of directly finite injective modules, we derive a surprisingly powerful technical result for quasi-continuous modules: the isomorphism type of the ‘internal hull’ of any submodule is determined by the isomorphism type of the submodule (Theorem 4).

CHAIN CONDITIONS ON QUOTIENT FINITE DIMENSIONAL MODULES

This paper is motivated by a recent work of C. Faith [9] who has proved that a quotient finite dimensional module which satisfies the ascending chain condition on subdirectly irreducible submodules is Noetherian. A natural question to ask (also raised by Faith [11]) is whether its dual holds true. We answer this in the affirmative, and provide a characterization of Artinian modules dual to Faiths Theorem. We also extend these results to the more general settings of dual Krull dimension and Krull dimension respectively.

Modules Whose Endomorphism Rings are Von Neumann Regular

Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of rings R are characterized in terms of endoregular R-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending an endoregular abelian group is also provided.

Principally Quasi-Baer Ring Hulls

We show the existence of principally (and finitely generated) right FI-extending right ring hulls for semiprime rings. From this result, we prove that right principally quasi-Baer (i.e., right p.q.-Baer) right ring hulls always exist for semiprime rings. This existence of right p.q.-Baer right ring hull for a semiprime ring unifies the result by Burgess and Raphael on the existence of a closely related unique smallest overring for a von Neumann regular ring with bounded index and the result of Dobbs and Picavet showing the existence of a weak Baer envelope for a commutative semiprime ring. As applications, we illustrate the transference of certain properties between a semiprime ring and its right p.q.-Baer right ring hull, and we explicitly describe a structure theorem for the right p.q.-Baer right ring hull of a semiprime ring with only finitely many minimal prime ideals. The existence of PP right ring hulls for reduced rings is also obtained. Further application to ring extensions such as monoid rings, matrix, and triangular matrix rings are investigated. Moreover, examples and counterexamples are provided.

MODULES WITH FI-EXTENDING HULLS

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension HFI(PR) (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved thatFI(End(PR)) ∼ End(HFI(PR)), where �FI(End(PR)) is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR)). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C ∗ -algebras is considered. Various examples which illustrate and delimit the results of this paper are provided. 2000 Mathematics Subject Classification. Primary 16N60, 16D40; Secondary 16S50.

An Essential Extension with Nonisomorphic Ring Structures II

We construct a ring R with R = Q(R), the maximal right ring of quotients of R, and a right R-module essential extension S R of R R such that S has several distinct isomorphism classes of compatible ring structures. It is shown that under one class of these compatible ring structures, the ring S is not a QF-ring (in fact S is not even a right FI-extending ring), while under all other remaining classes of the ring structures, the ring S is QF. We demonstrate our results by an application to a finite ring.