Essential submodules relative to a submodule

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Rajaee S.

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Journal Paper

Paper Information

Journal: Journal of Algebra and Related Topics Year:2023 | Volume:11 | Issue:2 Page(s): 59-71

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Information Journal Paper

Title

Essential submodules relative to a submodule

Author(s)

Rajaee S. | Issue Writer Certificate

Keywords

‎multiplication module

faithful module‎

‎essential submodule

Abstract

‎In this paper‎, ‎our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$‎. ‎Let $T$ be an arbitrary submodule of an $R$-module $M$‎, ‎then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎whenever for every submodule $X$ of $M$‎, ‎$N\cap X\subseteq T$ implies that‎ ‎$(T:M)\subseteq ^{e}{\rm Ann}(X)$‎. ‎We investigate some new results concerning to this class of submodules‎. ‎Among various results we prove that for a faithful multiplication $R$-module $M$‎, ‎if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$‎. ‎The converse is true if $M$ is moreover a finitely generated module‎.

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