By an essential product of two rings is meant a subdirect product which contains an
essential right ideal of the direct product. The aim of this paper is to investigate the utility of …

A ring is said to be a left essential extension of a reduced ring (domain) if it contains a left
ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided …

A (right nonsingular) ring R is called a splitting ring if, for every right R-module M, the
singular submodule Z (M) is a direct summand of M. If R is a semiprime splitting ring with …

Introduction: All rings in this paper are associative and have an identity. All modules will be
daitary. Let R be any ring and M a left R-module. A submodule N of M is called essential if …

Let R be a ring with identity. A module M_R MR is called an r-semisimple module if for any
right ideal I of R, MI is a direct summand of M_R MR which is a generalization of semisimple …

In a recent paper by Wang and Ding, it was stated that any ring which is generalized
supplemented as a left module over itself is semiperfect. The purpose of this note is to show …

The submodules with the property of the title (N⊆ M is strongly essential in M if∏ IN is
essential in∏ IM for any index set I) are introduced and fully investigated. It is shown that for …

All rings are assumed to be associative and to have a nonzero identity element. Let A be a
ring, and let M be a right A-module. A proper ideal P of the ring A is said to be right primitive …

An extension ring S of a ring T is called a left quotient ring of T if for any two elements ΛH= 0
and y of S there exists an element a of T such that ax^ rO and ay belongs to T. Let R be a …

This paper deals with the relationship between a ring T and the idealizer R of a right ideal M
of T.[The ring R is the largest subring of T which contains M as a two-sided ideal.] Assuming …