‏A ‎generalization ‎of‎ ‎‎r‎-submodules ‎

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Rajaee S.

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Journal Paper

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Journal: Journal of Algebra and Related Topics Year:2024 | Volume:12 | Issue:2 Page(s): 165-178

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Title

‏A ‎generalization ‎of‎ ‎‎r‎-submodules ‎

Author(s)

Rajaee S. | Issue Writer Certificate

Keywords

r‎-submodule

I-submodule

sr-submodule

special I-submodule

Abstract

‎Let $R$ be a commutative ring with identity $1\neq 0$ and $M$ a non-zero unital $R$-module‎. ‎In this paper‎, ‎we present the concept of fully $I$-submodules of $M$ such that $I$ is an ideal of $R$ which is a generalization of $r$-submodules‎. ‎Consider that $I$ is an ideal of $R$‎, ‎a proper submodule $N$ of $M$ is a fully $I$-submodule if $JK\subseteq N$ with ${\rm ann}_{M}(J)=0_{M}$ results that $K\subseteq N$ for each submodule $K$ of $M$ and each ideal $J$ of $R$‎. ‎In addition‎, ‎we present the concept of fully special $I$-submodules which is a generalization of special $r$-submodules‎. ‎A proper submodule $N$ of $M$ is a fully special $I$-submodule if the inclusion $IL\subseteq N$ with ${\rm ann}_{R}(L) = 0_{R}$‎, ‎implies that $I\subseteq (N:M)$ for each submodule $L$ of $M$ and each ideal $J$ of $R$‎. ‎We explore certain outcomes related to these categories of submodules‎.

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