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

Authors : Morteza NOFERESTİ, Hosein FAZAELİ MOGHİMİ, Mohammad Hossein HOSSEİNİ

Pages : 243-254

Doi:10.15672/hujms.605105

View : 16 | Download : 4

Publication Date : 2021-02-04

Article Type : Research Paper

Abstract :Let $\mathfrak{S}_pinsert ignore into journalissuearticles values(_RM);$ 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 $\eta:\mathfrak{S}_pinsert ignore into journalissuearticles values(_RR);\rightarrow \mathfrak{S}_pinsert ignore into journalissuearticles values(_RM);$ defined by $\etainsert ignore into journalissuearticles values(I);=S_pinsert ignore into journalissuearticles values(IM);$ and $\theta:\mathfrak{S}_pinsert ignore into journalissuearticles values(_RM);\rightarrow \mathfrak{S}_pinsert ignore into journalissuearticles values(_RR);$ defined by $\thetainsert ignore into journalissuearticles values(N);=insert ignore into journalissuearticles values(N:M);$, 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 $\mathfrak{S}_pinsert ignore into journalissuearticles values(_RM);$ forms a Boolean algebra and $\eta$ is a Boolean algebra homomorphism.
Keywords : Saturated submodule with respect to a prime ideal, eta module, heta module, mathfrak S distributive module, semisimple ring