Abstract
Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called s- closed submodule denoted by D ≤sc W, if D has no proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In this paper, we study modules which satisfies the ascending chain conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.
Highlights
Throughout this paper, L represents a commutative ring with unity and W be a left unitaryL- module
It s well known that “a submodule D of W is called small denoted by D
As a generalization of essential submodules, in [4] “Zhou and Zhang” introduced the concept of s- essential submodule, where “a submodule D of an L-module W is said to be an s -essential submodule of W denoted by D≤se W if D∩H=0 with H is a small submodule of W implies H= 0
Summary
Throughout this paper, L represents a commutative ring with unity and W be a left unitaryL- module. A submodule D of an L -module W is called s-closed submodule denoted by D ≤sc W, if D has no proper s -essential extension in W, that is , whenever D ≤W such that D ≤se K ≤ W, D = K. 11 : Let W be an L-module such that the s-essential submodules satisfy transitive property. Every noetherian (respectively artinian) module satisfies ACC ( respectively DCC ) on s -closed submodules.
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