MAPPINGS BETWEEN LATTICES OF RADICAL SUBMODULES

Year 2016, Volume: 19 Issue: 19 , 35 - 48 , 01.06.2016

Hosein Fazaeli Moghimi Javad Bagheri Harehdashti

https://doi.org/10.24330/ieja.266191

Cited By: 3

https://izlik.org/JA23EY47FE

Abstract

Let R be a ring and R(M) be the lattice of radical submodules
of an R-module M. Although the mapping ρ : R(R) → R(M) defined by
ρ(I) = rad(IM) is a lattice homomorphism, the mapping σ : R(M) → R(R)
defined by σ(N) = (N : M) is not necessarily so. In this paper, we examine
the properties of σ, in particular considering when it is a homomorphism. We
prove that a finitely generated R-module M is a multiplication module if and
only if σ is a homomorphism. In particular, a finitely generated module M
over a domain R is a faithful multiplication module if and only if σ is an
isomorphism.

Keywords

σ-modules , multiplication modules , primeful modules