The Unique Maximal GF‐Regular Submodule of a Module

Abstract

An R-module A is called GF-regular if, for each a ∈ A and r ∈ R, there exist t ∈ R and a positive integer n such that r n tr n a = r n a. We proved that each unitary R-module A contains a unique maximal GF-regular submodule, which we denoted by M GF(A). Furthermore, the radical properties of A are investigated; we proved that if A is an R-module and K is a submodule of A, then M GF(K) = K∩M GF(A). Moreover, if A is projective, then M GF(A) is a G-pure submodule of A and M GF(A) = M(R) · A.

Trivial Extension of π-Regular Rings

“…It is worth to mention that there are many generalizations of the concept of -regular rings to modules such as -reguar modules [6,7] and -regular modules [8] and there is an equivalent concept of -regular rings studied in [8] named -semiregular rings. And certainly there are many related concepts to -regular rings most notably -McCoy rings [9] and other related concepts as in [10].…”