Homological properties of 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 first study the transfer of some rings such as left [Formula: see text]-rings, left Kasch rings, left PS rings between [Formula: see text] and [Formula: see text]. Then we investigate how to construct [Formula: see text]-tilting modules and Gorenstein modules in [Formula: see text]-Mod from [Formula: see text]-Mod. Finally we exhibit the connections between preenvelopes (respectively, precovers) in [Formula: see text]-Mod and preenvelopes (respectively, precovers) in [Formula: see text]-Mod.