Relative Transpose and its Dual with Respect to a Bimodule

For a class of modules \(\mathcal {X}\), we introduce the \(\mathcal {X}\)-transpose of a module with respect to a bimodule, which unifies some well-known transposes. Let \(\mathcal {V}\) be a subclass of \(\mathcal {X}\). The relations between \(\mathcal {X}\)-transposes and \(\mathcal {V}\)-transposes are investigated under the condition that \(\mathcal {V}\) is a generator or cogenerator of \(\mathcal {X}\). The dual aspects of \(\mathcal {X}\)-transposes are also discussed. Then we give some applications of these results. In particular, the dual counterparts of Gorenstein transposes are established.