How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

Abstract: Let $\Delta =\left(\begin{smallmatrix} A & {AN_B}\ {_BM_A} & B \\end{smallmatrix}\right)$ be a Morita ring with $M\otimes{A}N=0=N\otimes_{B}M$.We first study how to construct (complete) duality pairs of $\Delta$-modules using (complete) duality pairs of $A$-modules and $B$-modules, generalizing the result of Mao (Comm. Algebra, 2020, 12: 5296--5310) about the duality pairs over a triangular matrix ring. Moreover, we construct Gorenstein projective modules relative to complete duality pairs of $\Delta$-modules. Finally, we give an application to Ding projective modules.