Relative transpose and its dual with respect to a bimodule (Q1697178)

Given a class \(\mathcal{X}\) of modules, the author introduces the notion of \(\mathcal{X}\)-transposes of modules with respect to a bimodule in the present paper, which unifies some well-known notions of transposes, such as Auslander transposes, Gorenstein transposes, \(C\)-Gorenstein transposes and so on. Several known results are extended, and for a subclass \(\mathcal{V}\) of \(\mathcal{X}\), some relations between \(\mathcal{X}\)-transposes and \(\mathcal{V}\)-transposes are established. The dual aspects of \(\mathcal{X}\)-transposes are also discussed in the paper. That is, the author introduces \(\mathcal{X}\)-cotransposes of modules with respect to a bimodule. In particular, the dual counterparts of Gorenstein transposes are established.