On serial rings

Let A be a ring such that all maximal indecomposable factor rings $A_i$ of $A$ are serial rings. Then every square matrix over $A$ is diagonalizable. In addition, if all the rings $A_i$ are Bezout rings, then every rectangular matrix over $A$ is diagonalizable. If $\varphi$ is an automorphism of the ring $A$, then the skew Laurent series ring $A((x,\varphi ))$ is a serial ring if and only if $A$ is a serial Artinian ring.