Let $ R $ and $ S $ be rings with unit. Suppose $ P $ is a free $ R $-module on $\beta $
generators, where $\beta $ is an infinite cardinal number not smaller than the cardinality of …

A submodule N of a module M is called S-closed (in M) if M/N is nonsingular. It is well-known
that the class Closed of short exact sequences determined by closed submodules is a …

Let X be a class of left R-modules. It is proved that if type 2, t-extending Ieft-R-modules Mr
and M2 are relatively essentially. te-injective and Ml is pseudoly M2-X'-injective (or M2 is …

A module M is called “self-small” if the functor Hom (M,−) commutes with direct sums of
copies of M. The main goal of the present article is to construct a non-self-small product of …

The study of pure-injectivity is accessed from an alternative point of view. A module M is
called pure-subinjective relative to a module N if for every pure extension K of N, every …

Abstract A left R-module M is said to be fp-injective if, for every monomorphism K→ L with K
and L finitely presented left R-modules, Hom (L, M)→ Hom (K, M) is an epimorphism. A right …

Let E be an injectively resolving subcategory of le R-modules. A le R-module M (resp. right
R-module N) is called E-injective (resp. E-at) if ExtR (G, M)=(resp. TorR (N, G)=) for any G∈ …

Let R be an arbitrary ring with identity and M, a right R-module with Z 2 (M) the second
singular submodule of M. We call an endomrphism φ of M, a t-automorphism if φ ̄: M∕ Z 2 …

This paper is devoted to the study of the endo-structure of infinite direct sums $\bigoplus_
{i\in I} M_i $ of indecomposable modules $ M_i $ over a ring $ R $. It is centered on the …

Let R be a right pure semisimple ring, and be a family of sources of left almost split
morphisms in mod-R. We study the definable subcategory in Mod-R determined by the …