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Strongly Gorenstein flat modules. (English) Zbl 1200.16010

Summary: Strongly Gorenstein flat modules are introduced and investigated. An \(R\)-module \(M\) is called strongly Gorenstein flat if there is an exact sequence \(\cdots\to P_1\to P_0\to P^0\to P^1\to\cdots\) of projective \(R\)-modules with \(M=\ker(P^0\to P^1)\) such that \(\operatorname{Hom}(-,F)\) leaves the sequence exact whenever \(F\) is a flat \(R\)-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

MSC:

16E05 Syzygies, resolutions, complexes in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
16E10 Homological dimension in associative algebras
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
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References:

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