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On semisimple extensions of serial rings

We prove that if B is a commutative local serial ring and A is a B-algebra which is a left semisimple extension of B , A is a uniserial ring. If in addition A is indecomposable as ring, the lengths of the composition serieses of Ae and B are same for each primitive idempotent e of A . We also give some necessary and sufficient conditions for A to be a left semisimple extension of a subring B of it, in the case where A and B are local serial rings or the case where B is a commutative local serial ring and A is a B-algebra which is serial.