Modules are S-modules where S is an arbitrary ring with or without a unit element. We
consider a projective module P having a submodule K such that K+ Y= P implies that the …

0) where we assume that f2 is surjective. Then (1) is called a Cartesian square and R a
pullback or fibre product of Rx and R2 over R', if gx is surjective and g2 induces an …

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 …

An R-module M is called∑-extending if every coproduct of copies of M is extending, ie
closed submodules are direct summands. Oshiro (1984) has shown that the ring R is∑ …

In [5], Roiter has solved the Brauer-Thrall conjecture for finite-dimensiona algebras over
fields, which states that, if the lengths of the finitely generated indecomposable modules are …

Let D be an R-module over an arbitrary ring R of projective dimension at most 1. We
construct an R-module G containing D such that Ext (D, G)= 0= Ext (G, G). Moreover, we …

Classically, there has been, for obvious reasons, an intimate relation between the concepts"
rings of quotients" and" divisible modules". Recently, however, their generalizations have …

The definition of the determinant of an endomorphism of a free module depends on the
following fact: If F is a free R-module of rank n, then the homogeneous component∧ nF, of …

\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 …

A characterization is given of the finitely generated nonsingular left R-modules N such that
Ext R 1 (N, M)= 0 for every singular left R-module M. As a corollary, the rings R, over which …