Mappings between the lattices of saturated submodules with respect to a prime ideal

Year 2021, Volume: 50 Issue: 1, 243 - 254, 04.02.2021

Morteza Noferesti Hosein Fazaeli Moghimi , Mohammad Hossein Hosseini

https://doi.org/10.15672/hujms.605105

Cited By: 1

Abstract

Let $\mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an $R$-module $M$ with respect to a prime ideal $p$ of a commutative ring $R$. We examine the properties of the mappings $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$, in particular considering when these mappings are lattice homomorphisms. It is proved that if $M$ is a semisimple module or a projective module, then $\eta$ is a lattice homomorphism. Also, if $M$ is a faithful multiplication $R$-module, then $\eta$ is a lattice epimorphism. In particular, if $M$ is a finitely generated faithful multiplication $R$-module, then $\eta$ is a lattice isomorphism and its inverse is $\theta$. It is shown that if $M$ is a distributive module over a semisimple ring $R$, then the lattice $\mathfrak{S}_p(_RM)$ forms a Boolean algebra and $\eta$ is a Boolean algebra homomorphism.

Keywords

Saturated submodule with respect to a prime ideal, $\eta$-module, $\theta$-module, $\mathfrak{S}$-distributive module, semisimple ring

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