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Foxby equivalence over associative rings. (English) Zbl 1154.16007

The notion of a dualizing module over a commutative, Noetherian ring has been extended to that of a semidualizing module. These modules have interesting connections with the Auslander and Bass classes of modules (notions introduced by Foxby) and with Foxby equivalence between these classes and several naturally defined subclasses of them.
In this paper the authors extend the notion of a semidualizing module to that of a semidualizing bimodule over a pair \((S,R)\) of not necessarily commutative rings. Then they show the various versions of Foxby equivalence hold in this general situation and also give results which are new even in the original setting.
This clearly written article can serve as a pleasant introduction to this interesting area of homological algebra.

MSC:

16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16D20 Bimodules in associative algebras
16E10 Homological dimension in associative algebras
18G99 Homological algebra in category theory, derived categories and functors
16D90 Module categories in associative algebras
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