On a generalization of $z$-ideals in modules over commutative rings
In this article, we introduce and study the concept of $z$-submodules as a generalization of $z$-ideals. Let $M$ be a module over a commutative ring with identity $R$. A proper submodule $N$ of $M$ is called a $z$-submodule if for any $x\in M$ and $y\in N$ such that every maximal submodule of $M$ co...
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| Veröffentlicht in: | International electronic journal of algebra Jg. 37; H. 37; S. 297 - 312 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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14.01.2025
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| ISSN: | 1306-6048, 1306-6048 |
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| Abstract | In this article, we introduce and study the concept of $z$-submodules as a generalization of $z$-ideals. Let $M$ be a module over a commutative ring with identity $R$. A proper submodule $N$ of $M$ is called a $z$-submodule if for any $x\in M$ and $y\in N$ such that every maximal submodule of $M$ containing $y$ also contains $x$, then $x\in N$ as well. We investigate the properties of $z$-submodules, particularly considering their stability with respect to various module constructions. Let $\mathcal{Z}({_R}M)$ denote the lattice of $z$-submodules of $M$ ordered by inclusion. We are concerned with
certain mappings between the lattices $\mathcal{Z}({_R}R)$ and $\mathcal{Z}({_R}M)$. The mappings in question are $\phi:\mathcal{Z}({_R}R) \rightarrow \mathcal{Z}({_R}M)$ defined by setting for each $z$-ideal $I$ of $R$, $\phi(I)$ to be the intersection of all $z$-submodules of $M$ containing $IM$ and $\psi:\mathcal{Z}({_R}M) \rightarrow \mathcal{Z}({_R}R)$ defined by $\psi(N)$ is the colon ideal $(N:M)$. It is shown that $\phi$ is a lattice homomorphism, and if $M$ is a finitely generated multiplication module, then $\psi$ is also a lattice homomorphism. In particular, $\mathcal{Z}({_R}M)$ is a homomorphic image of $\mathcal{R}({_R}M)$, the lattice of radical submodules of $M$. Finally, we show that if $Y$ is a finite subset of a compact Hausdorff $P$-space $X$, then every submodule of the $C(X)$- module $\mathbb{R}^Y$ is a $z$-submodule of $\mathbb{R}^Y$. |
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| AbstractList | In this article, we introduce and study the concept of $z$-submodules as a generalization of $z$-ideals. Let $M$ be a module over a commutative ring with identity $R$. A proper submodule $N$ of $M$ is called a $z$-submodule if for any $x\in M$ and $y\in N$ such that every maximal submodule of $M$ containing $y$ also contains $x$, then $x\in N$ as well. We investigate the properties of $z$-submodules, particularly considering their stability with respect to various module constructions. Let $\mathcal{Z}({_R}M)$ denote the lattice of $z$-submodules of $M$ ordered by inclusion. We are concerned with
certain mappings between the lattices $\mathcal{Z}({_R}R)$ and $\mathcal{Z}({_R}M)$. The mappings in question are $\phi:\mathcal{Z}({_R}R) \rightarrow \mathcal{Z}({_R}M)$ defined by setting for each $z$-ideal $I$ of $R$, $\phi(I)$ to be the intersection of all $z$-submodules of $M$ containing $IM$ and $\psi:\mathcal{Z}({_R}M) \rightarrow \mathcal{Z}({_R}R)$ defined by $\psi(N)$ is the colon ideal $(N:M)$. It is shown that $\phi$ is a lattice homomorphism, and if $M$ is a finitely generated multiplication module, then $\psi$ is also a lattice homomorphism. In particular, $\mathcal{Z}({_R}M)$ is a homomorphic image of $\mathcal{R}({_R}M)$, the lattice of radical submodules of $M$. Finally, we show that if $Y$ is a finite subset of a compact Hausdorff $P$-space $X$, then every submodule of the $C(X)$- module $\mathbb{R}^Y$ is a $z$-submodule of $\mathbb{R}^Y$. |
| Author | Mohebian, Seyedeh Fatemeh Fazaeli Moghimi, Hosein |
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| Cites_doi | 10.1007/s41980-021-00573-z 10.1007/BF01140126 10.15672/hujms.605105 10.1007/978-1-4615-7819-2 10.1016/0021-8693(73)90024-0 10.24330/ieja.266191 10.2989/16073606.2019.1601647 10.1216/JCA-2015-7-4-567 10.24330/ieja.266224 10.15672/hujms.455030 10.4064/fm-45-1-28-50 10.1080/00927878808823601 10.1081/AGB-120014684 10.1016/j.topol.2019.106969 10.4153/CMB-1980-064-3 10.24330/ieja.266246 10.7146/math.scand.a-11411 10.2307/2040685 10.1515/ms-2017-0099 |
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