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On \(n\)-copresented modules and \(n\)-co-coherent rings. (English) Zbl 1253.13020

This paper develops the concepts of \(N\)-copresented modules and \(n\)-co-coherent rings as duals to the notions of \(N\)-presented modules and \(n\)-coherent rings along the lines of the development of finitely copresented modules and co-coherent rings as duals to finitely presented modules and coherent rings. Let \(R\) be a commutative ring and \(M\) an \(R\)-module. \(M\) is defined to be \(n\)-copresented if there exists an exact sequence of \(R\)-modules \(0 \rightarrow M \rightarrow I_{0} \rightarrow I_{1} \rightarrow \ldots \rightarrow I_{n}\), with \(I_{i}\) injective and finitely cogenerated for \(i= 0, \ldots , n.\) \(M\) is called infinitely copresented if \(M\) is \(n\)-copresented for every natural number \(n\). It is shown that 0-copresentedness (resp. 1-co-presentedness) coincides with finitely cogeneratedness (resp. finitely copresentedness). Properties of \(n\)-copresented modules are investigated, in particular how \(n\)-copresentedness of modules in a short exact sequence relates to one another and some change of ring properties are also studied.
A ring \(R\) is called \(n\)-co-coherent (\(n\) any positive integer) if every \(n\)-copresented \(R\)-module is \((n+1)\)-copresented. 0-co-coherent rings are shown to be identical to co-Noetherian rings and 1-co-coherent rings coincide with co-coherent rings. If \(R\) is an \(n\)-co-coherent semi-local ring, then \(R\) is shown to be \(n\)-coherent. The paper concludes with a theorem showing that a direct product of rings is \(n\)-co-coherent if and only if each component is \(n\)-co-coherent.

MSC:

13E15 Commutative rings and modules of finite generation or presentation; number of generators
13E99 Chain conditions, finiteness conditions in commutative ring theory
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