Homological aspects of n -trivial extensions of rings

Let [Formula: see text] be the [Formula: see text]-trivial extension of a ring [Formula: see text] by an [Formula: see text]-bimodule [Formula: see text]. We first characterize projective, injective, flat and finitely generated modules over [Formula: see text]. Then we describe when the ring [Formula: see text] is left perfect (Noetherian, coherent, Artinian, hereditary, Kasch). Finally, we establish some homological formulas of [Formula: see text]-trivial extensions of rings.