On semisimple extensions of serial rings (Q1977492)

A ring \(A\) is said to be a left semisimple extension of a unital subring \(B\) if the multiplication map \(A\otimes_BM\to M\) splits as a left \(A\)-homomorphism for every \(_RM\). If \(B\) is a commutative local serial ring, \(A\) is a \(B\)-algebra, and \(A\) is a left semisimple extension of \(B\), then \(A\) is a uniserial ring; moreover, if \(A\) is indecomposable as a ring, then the length of the composition series of \(Ae\) coincides with that of \(B\) for each primitive idempotent \(e\) of \(A\). Let \(N\) denote the radical of \(A\), and let \(J\) denote the radical of \(B\). If \(B\) is a commutative local serial ring and \(A\) is a semiprimary \(B\)-algebra such that \(N=AJ\), then \(A\) is a uniserial ring; moreover; if \(A\) is a serial \(B\)-algebra, then \(A\) is a left semisimple extension of \(B\) if and only if \(N=JA\). The following conditions are equivalent for local serial rings \(A\) and \(B\): (1) \(A\) is a left semisimple extension of \(B\), (2) \(N=AJ\), (3) the lengths of the composition series of \(_AA\) and \(_BB\) are the same. If a local ring \(A\) is a left semisimple extension of a local serial ring \(B\) and \(_BA\) is finitely generated, then \(A\) is a left serial ring. Examples are constructed to illustrate the theory.