Let R be a ring with Jacobson radical J (R). Following CNI, a submodule N of a left R-
module RM is said to lie over a summand of M if there exists an idempotent e E End (RM) …

The object of this paper is to study the relationship between certain projective modules and
their endomorphism rings. Specifically, the basic problem is to describe the projective …

The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the
behavior of the objects of cocycles in acyclic complexes. It is known that any flat Proj …

Let $ R $ be a ring and $ E= E (R_R) $ its injective envelope. We show that if every simple
right $ R $-module embeds in $ R_R $ and every cyclic submodule of $ E_R $ is essentially …

This paper studies infinite acyclic complexes of finitely generated free modules over a
commutative noetherian local ring $(R,\fm) $ with $\fm^ 3= 0$. Conclusive results are …

Given a module M over a commutative ring R, it is of considerable interest to obtain
information on the homological properties of its symmetric powers SiM and and exterior …

Publisher Summary This chapter provides an overview of the basic concepts of the Krull-
Schmidt theorem. The two proofs of the Krull-Schmidt-Azumaya theorem, mention …

In this paper, we study interplay between exchange ring R and their J-semisimple
indecomposable factor rings. In particular, we prove that for any (exchange) ring R and any …

T (M)=(x= M| Hom,(Ra, I (R))= 0) where I (R) is the injective hull of R as a left R-module. A left
R-module M is said to be torsion free if T (M)= 0. For the properties of torsion free modules …

This unique and comprehensive volume provides an up-to-date account of the literature on
the subject of determining the structure of rings over which cyclic modules or proper cyclic …