Morita equivalence and quotient rings

Abstract

Let A and B be rings and let M be an A–B-bimodule that is finitely generated and projective in A-mod and in mod-B. Also let I be an ideal of A and let J be an ideal of B such that IM=MJ. Our main result is a partial converse of a known result:Proposition.Suppose thatI≤J(A),J≤J(B)so thatM/(IM)is anA‾=A/I–B‾=B/J-bimodule that is finitely generated and projective inA‾-mod and in mod-B‾and that induces a Morita Equivalence betweenA‾-mod andB‾-mod. Then M induces a Morita Equivalence between A-mod and B-mod.This result should be particularly useful in the context that A and B are O-algebras where O is a commutative local ring, I=J(O)A and J=I(O)B. In which case, A‾ and B‾ are finite dimensional algebras over the field k=O/J(O).