On the distributivity of the lattice of radical submodules

Document Type : Research Paper

Authors

Department of Mathematics, University of Birjand, Birjand, Iran

Abstract

Let $R$ be a commutative ring with identity and $\mathcal{R}(_{R}M)$ denotes the bounded lattice of radical submodules of an $R$-module $M$. We say that $M$ is a radical distributive module, if $\mathcal{R}(_{R}M)$ is a distributive lattice. It is shown that the class of radical distributive modules contains the classes of multiplication modules and finitely generated distributive modules properly. It is shown that if $M$ is a semisimple $R$-module and for any radical submodule $N$ of $M$ with direct sum complement $\tilde{N}$, the complementary operation on $\mathcal{R}(_{R}M)$ is defined by $N':=\tilde{N}+rad(0)$, then $\mathcal{R}(_{R}M)$ with this unary operation forms a Boolean algebra. In particular, if $M$ is a multiplication module over a semisimple ring $R$, then $\mathcal{R}(_{R}M)$ is a Boolean algebra, which is also a homomorphic image of $\mathcal{R}(_{R}R)$.

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References

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Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 347-355

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History

  • Receive Date: 16 May 2023
  • Revise Date: 20 July 2023
  • Accept Date: 29 September 2023

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