LAUSR

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