Cosmin S. Roman

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.

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.

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.

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M R is a finitely generated 𝔏-Rickart module, we prove that End R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End R (M) is a left extending ring iff M is a Baer module and End R (M) is left cononsingular.

Modules Whose Endomorphism Rings are Division Rings

The well-known Schurs Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.