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Generalized coherent rings by Gorenstein projective dimension. (English) Zbl 1187.16005

Summary: We introduce a new generalization of coherent rings using the Gorenstein projective dimension. Let \(n\) be a positive integer or \(n=\infty\). A ring \(R\) is called a left \(G_n\)-coherent ring in case every finitely generated submodule of finitely generated free left \(R\)-modules whose Gorenstein projective dimension \(\leq n-1\) is finitely presented. We characterize \(G_n\)-coherent rings in various ways, using \(G_n\)-flat and \(G_n\)-injective modules and cotorsion theory.

MSC:

16E10 Homological dimension in associative algebras
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)