Open AccessA New Characterization of Commutative Strongly $\Pi$-Regular Rings
Author(s) -
Anta Niane Gueye,
Cheikh Thiécoumba Gueye,
Mamadou Sanghare
Publication year - 2012
Publication title -
journal of mathematics research
Language(s) - English
Resource type - Journals
eISSN - 1916-9809
pISSN - 1916-9795
DOI - 10.5539/jmr.v4n5p30
Subject(s) - mathematics , characterization (materials science) , commutative ring , maximal ideal , commutative property , von neumann regular ring , isomorphism (crystallography) , finitely generated abelian group , prime (order theory) , injective function , prime ideal , ideal (ethics) , pure mathematics , endomorphism , primary ideal , ring (chemistry) , discrete mathematics , principal ideal ring , combinatorics , crystallography , chemistry , materials science , crystal structure , organic chemistry , nanotechnology , philosophy , epistemology
Let $R$ be a commutative ring. It is known that any injective endomorphism of finitely generated $R$-module is an isomorphism if and only if every prime ideal of $R$ is maximal. This result makes possible a characterization of rings on which all finitely generated modules are co-hopfian. The motivation of this paper comes from trying to extend these results to mono-correct modules. In doing so, we show that any finitely generated $R$-module is mono-correct if and only if every prime ideal of $R$ is maximal and we obtain a characterization of commutative rings on which all finitely generated module are mono-correct. Such rings are exactly commutative strongly $Pi$-regular rings. So we have a new characterization of commutative strongly $Pi$-regular rings