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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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
14.01.2025
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| ISSN: | 1306-6048, 1306-6048 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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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| ISSN: | 1306-6048 1306-6048 |
| DOI: | 10.24330/ieja.1555106 |