| On $(m,n)$-Coherent Modules and Preenvelopes |
| Received:January 03, 2006 Revised:August 26, 2006 |
| Key Words:
$(m,n)$-$M$-flat module $(m,n)$-coherent module $(m,n)$-$M$-flat preenvelope.
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| Fund Project:the National Natural Science Foundation of China (No. 10571026); the Natural Science Foundation of Anhui Provincial Education Department (No. 2006kj050c); Doctoral Foundation of Anhui Normal University. |
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| Abstract: |
| In this paper, let $m,n$ be two fixed positive integers and $M$ be a right $R$-module, we define $(m,n)$-$M$-flat modules and $(m,n)$-coherent modules. A right $R$-module $F$ is called $(m,n)$-$M$-flat if every homomorphism from an $(n,m)$-presented right $R$-module into $F$ factors through a module in ${\rm add}M$. A left $S$-module $M$ is called an $(m,n)$-coherent module if $M_{R}$ is finitely presented, and for any $(n,m)$-presented right $R$-module $K$, ${\rm Hom}(K,M)$ is a finitely generated left $S$-module, where $S={\rm End}(M_{R})$. We mainly characterize $(m,n)$-coherent modules in terms of preenvelopes (which are monomorphism or epimorphism) of modules. Some properties of $(m,n)$-coherent rings and coherent rings are obtained as corollaries. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.01.009 |
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