Mappings between the lattices of varieties of submodules

Document Type : Research Paper

Authors

Department of Mathematics, University of Birjand, Birjand, Iran.

Abstract

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.

Keywords

Journal of Algebra and Related Topics
Volume 10, Issue 1
June 2022
Pages 35-50

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