-Regular Modules

Abstract

We introduced and studied -regular modules as a generalization of -regular rings to modules as well as regular modules (in the sense of Fieldhouse). An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff is a -regular -module for each maximal ideal of . Many characterizations and properties of -regular modules were given. An -module is -regular iff is a -regular ring for each iff is a -regular ring for finitely generated module . If is a -regular module, then .

The Unique Maximal GF‐Regular Submodule of a Module

“…Now, by Zorn's lemma, G contains a maximal element which we call M GF . To prove the uniqueness of M GF , assume that M GF 1 and M GF 2 be two maximal GF -regular submodules in A ; then for any maximal ideal P of R each of M GF 1 p and M GF 2 p is semisimple over R p [2, Proposition 21]. Now, let M GF 1 p ∩ M GF 2 p = K p ; then K p ⊆ M GF 1 p and K p ⊆ M GF 2 p ; thus M GF 1 p = K p + A 1 p and M GF 2 p = K p + A 2 p , where A 1 p and A 2 p are two submodules of A p [4].…”

“…Now, let M GF 1 p ∩ M GF 2 p = K p ; then K p ⊆ M GF 1 p and K p ⊆ M GF 2 p ; thus M GF 1 p = K p + A 1 p and M GF 2 p = K p + A 2 p , where A 1 p and A 2 p are two submodules of A p [4]. Hence, M GF 1 p + M GF 2 p = A 1 p + K p + A 2 p , but each of A 1 p , A 2 p , and K p is a semisimple submodule; thus M GF 1 p + M GF 2 p is a semisimple submodule which implies that M GF 1 p + M GF 2 p is GF -regular [2]. So M GF 1 + M GF 2 is a GF -regular submodule [2, Theorem 20].…”

“…Hence, M GF 1 p + M GF 2 p = A 1 p + K p + A 2 p , but each of A 1 p , A 2 p , and K p is a semisimple submodule; thus M GF 1 p + M GF 2 p is a semisimple submodule which implies that M GF 1 p + M GF 2 p is GF -regular [2]. So M GF 1 + M GF 2 is a GF -regular submodule [2, Theorem 20]. Now, each of M GF 1 and M GF 2 is a maximal GF -regular submodule and hence M GF 1 + M GF 2 = M GF 2 = M GF 1.…”

“…The concept of regularity was extended to modules in several ways and in [2] the notion of F -regular modules (in the sense of Fieldhouse [3]) was generalized to GF -regular modules. Let A be an R -module; an element a ∈ A is said to be GF -regular if for each r ∈ R there exist t ∈ R and a positive integer n such that r n tr n a = r n a .…”

“…A submodule N of an R -module A is called GF -regular if each element of N is GF -regular and every submodule of a GF -regular module is a GF -regular module. Also, in [2] the concept of G -pure submodules was introduced; a submodule P of an R -module A is called G -pure if, for each r ∈ R , there exists a positive integer n such that P ∩ Rr n A = Rr n P .…”