Duality pairs over trivial ring extensions

Let [Formula: see text] be an associative ring with identity and [Formula: see text] a trivial extension of [Formula: see text] by an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text]. We first construct duality pairs of [Formula: see text]-modules by using duality pairs of [Formula: see text]-modules. Then we characterize some classes of Gorenstein projective modules relative to complete duality pairs of [Formula: see text]-modules.