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.
Rajaee, S. (2023). Essential submodules relative to a submodule. Journal of Algebra and Related Topics, 11(2), 59-71. doi: 10.22124/jart.2023.23331.1465
MLA
S. Rajaee. "Essential submodules relative to a submodule". Journal of Algebra and Related Topics, 11, 2, 2023, 59-71. doi: 10.22124/jart.2023.23331.1465
HARVARD
Rajaee, S. (2023). 'Essential submodules relative to a submodule', Journal of Algebra and Related Topics, 11(2), pp. 59-71. doi: 10.22124/jart.2023.23331.1465
VANCOUVER
Rajaee, S. Essential submodules relative to a submodule. Journal of Algebra and Related Topics, 2023; 11(2): 59-71. doi: 10.22124/jart.2023.23331.1465
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