Remark

It follows from the proof of Lemma 2 that M and N are $(Z,A)$-bimodules of finite length. Consequently, if B and A are two semisimple rings that are finitely generated modules over their respective centers Z(B) and Z(A), then two $(B$, A)-bimodules of finite length that are isomorphic as (Z(B),Z(A))-bimodules are isomorphic.