Publication

A note on prime modules

Bibliographic Details
Summary:	In this note we compare some notions of primeness for modules existing in the literature. We characterize the prime left R-modules such that the left annihilator of every element is a (two-sided) ideal of R, where R is an associative ring with unity, and we prove that if M is such a left R-module then M is strongly prime. These two notions are studied by Beachy [B75]. Furthermore, if M is projective as left R=(0 : M)-module then M is B-prime in the sense of Bican et al. [BJKN]. On the other hand, if M is faithful then M is (strongly) prime if and only if M is strongly prime (or an SP-module) in the sense of Handelman-Lawrence [HL] if and only if M is torsionfree and R is a domain. In particular this happens if R is commutative.
Subject:	Algebra, Mathematics Álgebra, Matemática
Country:	Portugal
Document type:	journal article
Access type:	Open
Associated institution:	Repositório Aberto da Universidade do Porto
Language:	English
Origin:	Repositório Aberto da Universidade do Porto

Description
Summary:	In this note we compare some notions of primeness for modules existing in the literature. We characterize the prime left R-modules such that the left annihilator of every element is a (two-sided) ideal of R, where R is an associative ring with unity, and we prove that if M is such a left R-module then M is strongly prime. These two notions are studied by Beachy [B75]. Furthermore, if M is projective as left R=(0 : M)-module then M is B-prime in the sense of Bican et al. [BJKN]. On the other hand, if M is faithful then M is (strongly) prime if and only if M is strongly prime (or an SP-module) in the sense of Handelman-Lawrence [HL] if and only if M is torsionfree and R is a domain. In particular this happens if R is commutative.