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Mappings between the lattices of varieties of submodules | ||
| Journal of Algebra and Related Topics | ||
| دوره 10، شماره 1، شهریور 2022، صفحه 35-50 اصل مقاله (304.64 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2021.19574.1272 | ||
| نویسندگان | ||
| H. Fazaeli Moghimi* ؛ M. Noferesti | ||
| Department of Mathematics, University of Birjand, Birjand, Iran. | ||
| چکیده | ||
| Let $R$ be a commutative ring with identity and $M$ be an $R$-module. It is shown that the usual lattice $\mathcal{V}(_{R}M)$ of varieties of submodules of $M$ is a distributive lattice. If $M$ is a semisimple $R$-module and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are lattice homomorphisms. It is shown that if $M$ is a faithful primeful $R$-module, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic lattices, and therefore $\mathcal{V}(_{R}M)$ and the lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras. | ||
| کلیدواژه‌ها | ||
| Lattice homomorphism؛ $omega$-module؛ Primeful module؛ Semisimple ring | ||
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آمار تعداد مشاهده مقاله: 583 تعداد دریافت فایل اصل مقاله: 560 |
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