On the distributivity of the lattice of radical submodules

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Fazaeli Moghimi Hossein; Noferesti Morteza

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Journal Paper

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Journal: Journal of Mahani Mathematical Research Year:2024 | Volume:13 | Issue:1 Page(s): 347-355

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Information Journal Paper

Title

On the distributivity of the lattice of radical submodules

Author(s)

Fazaeli Moghimi Hossein | Noferesti Morteza | Issue Writer Certificate

Keywords

Radical distributive module

distributive module

multiplication module

semisimple ring

Boolean algebra homomorphism

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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