Weakly Primary Submodules over Noncommutative Rings

Abstract

Let R be an associative ring with nonzero identity and let M be a unitary left R−module. In this paper, we introduce the concept of weakly primary submodules of M and give some basic properties of these classes of submodules. Several results on weakly primary submodules over noncommutative rings are proved. We show that N is a weakly primary submodule of a left R−module M iff for every ideal P of R and for every submodule D of M with 0 ̸= P D ⊆ N, either P ⊆ √ (N : M) or D ⊆ N. We also introduce the definitions of weakly primary compactly packed and maximal compactly packed modules. Then we study the relation between these modules and investigate the condition on a left R−module M that makes the concepts of primary compactly packed modules and weakly primary compactly packed modules equivalent. We also introduce the concept of weakly primary radical submodules and show that every Bezout module that satisfies the ascending chain condition on weakly primary radical submodules is weakly primary compactly packed module.