J. Korean Math. Soc. 2009; 46(1): 99-111
Printed January 1, 2009
Copyright © The Korean Mathematical Society.
Lixin Mao
Nanjing Institute of Technology
A ring is called left $P$-coherent if every principal left ideal is finitely presented. Let $M$ be a right $R$-module with the endomorphism ring $S$. We mainly study the $P$-coherence of $S$. It is shown that $S$ is a left $P$-coherent ring if and only if the left annihilator ann$_{S}(X)$ is a finitely generated left ideal of $S$ for any $M$-cyclic submodule $X$ of $M$ if and only if every cyclically $M$-presented right $R$-module has an $M$-torsionfree preenvelope. As applications, we investigate when the endomorphism ring $S$ is left $PP$ or von Neumann regular.
Keywords: $P$-coherent ring, $M$-torsionfree module, preenvelope
MSC numbers: 16P70, 16D20, 16D40
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