In § 1, we begin with a basic theorem which describes a convenient embedding of a
nonsingular left R-module into a complete direct product of copies of the left injective hull of …

1. Introduction. It is a well-known and basic result of homological alge-bra that the direct
product of an arbitrary family of injective modules over any ring is again injective [3, p. 8] …

In a celebrated theorem, AW Goldie determined necessary and sufficient conditions that a
ring have a left quotient ring which is semisimple with minimum condition [4]. Our first main …

A ring R is left noetherian if and only if the direct sum of injective envelopes of any family of
left R-modules is the injective envelope of the direct sum of the given family of modules (or …

1. Introduction. Let R be a ring (associative and with identity). It is easy to see that every left
ideal of R is the (left) annihilator of a finite subset of R if and only if every cyclic left R-module …

Introduction. The problem considered is a special case of the general problem of the extent
to which a module (F, A) is determined by its endomorphism ring E (F, A). We restrict our …

In [2 1, Fuchs has shown that R is a test module for quasiinjective modules. He showed that
if M is a right R-module, M is quasi-injective if and only if for each right ideal K of R, every R …

Goldie's torsion class 𝒢 is a class of left R-modules closed under taking submodules, factor
modules, extensions, arbitrary direct sums, and injective envelopes. The corresponding …

We characterize the left Noetherian rings by the existence of decompositions of left modules
into direct sums of an injective submodule and a submodule containing no injective …

In [2, page 151], J. Lambek proposes the following exercise: With any maximal right ideal M
of a ring R with 1 associate the ideal. Show that OM, is the right annihilator of the injective …