The purpose of this paper is to study the direct sum of indecomposable projective modules
which has the (finite) exchange property. A module M is said to have the exchange property …

This paper contains two main results. The first gives a necessary and sufficient condition for
a direct sum of indecomposable injective modules to have the exchange property. It is seen …

In this paper, a new class of rings, called exchange rings, is defined and studied. The class
of exchange rings includes regular rings in the sense of yon Neumann, local rings, and …

Let A be a ring with identity and let all modules be unital. The ring A is said to be of finite
representation type provided A is right artinian and has only finite number of finitely …

The classification of finitely generated projective modules over a ring A can be, as a rule,
divided into two rather different parts. The first, the classification up to stable isomorphism, is …

It is well known that all projective modules over a perfect ring have the exchange property.
This was shown by Yamagata [32] and, independently, by Harada and Ishii [lo]. Provided …

Introduction. This note investigates modules containing an essential direct sum of uniform
submodules. Theorem I gives five different necessary and sufficient conditions in order for a …

\S 0. Introduction. The main purpose of this paper is to extend some results on completely
reducible modules to quasi-injective modules by replacing “irreducible submodules” with …

In searching for isomorphic refinements of direct-sum decompositions of modules9 Crawley
and Jonsson introduced an exchange property for modules that is strongly reminiscent of …

It is an open problem to determine which ring satisfies the condition that every protective
module over the ring has the (finite) exchange property. For this problem, Harada and Ishii …