Finitely presented modules and phantom morphisms 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], and let [Formula: see text] be a right [Formula: see text]-module. It is proved that if [Formula: see text] is a finitely presented right [Formula: see text]-module, then [Formula: see text] is a finitely presented right [Formula: see text]-module; conversely, if [Formula: see text] is a finitely presented right [Formula: see text]-module, and the sequence [Formula: see text] is exact, then [Formula: see text] is a finitely presented right [Formula: see text]-module. This result is used to characterize when phantom (respectively, cophantom) morphisms in [Formula: see text]-[Formula: see text] can induce phantom (respectively, cophantom) morphisms in [Formula: see text]-[Formula: see text]. Let [Formula: see text] be a phantom morphism of [Formula: see text]-modules. If [Formula: see text] is a projective left [Formula: see text]-module, then it is shown that [Formula: see text] is a phantom morphism of [Formula: see text]-modules; dually, if [Formula: see text] is a flat left [Formula: see text]-module, and if [Formula: see text] is a cophantom morphism in [Formula: see text]-[Formula: see text], then [Formula: see text] is a cophantom morphism in [Formula: see text]-[Formula: see text].