Frobenius pairs over trivial ring extensions

Let [Formula: see text] be a trivial extension of a ring [Formula: see text] by an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text]. We investigate how to construct Frobenius pairs (respectively, strong Frobenius pairs) over [Formula: see text] by using Frobenius pairs (respectively, strong Frobenius pairs) over [Formula: see text]. As an application, we apply some of our results to Morita rings with zero bimodule homomorphisms.