SOCLE SERIES OF A COMMUTATIVE ARTINIAN RING

Abstract

Let $R$ be a commutative artinian ring, and $f(x)\in R[x]$ be a non-constant monic polynomial. The main purpose of this paper is to determine the socle series of $R[x]/\langle f(x)\rangle$ in terms of the socle series of $R$. As an application of the results proved, it is proved that $R$ is a $QF$-ring if and only if $R[x]/\langle f(x)\rangle$ is a $QF$-ring. As another application , a necessary and sufficient condition for a local artinian ring $R$ having a semisimple ideal $B$, with $R/B$ a $PIR$, to be a split extension of a $PIR$ by a semisimple module, is given.