## A representation theorem for Chain rings

### Volume 96 / 2003

Colloquium Mathematicum 96 (2003), 103-119
MSC: Primary 16S35; Secondary 16P10, 16P20.
DOI: 10.4064/cm96-1-10

#### Abstract

A ring $A$ is called a *chain ring* if it is a local, both sided artinian, principal ideal ring. Let $R$ be a commutative chain ring. Let $A$ be a faithful $R$-algebra which is a chain ring such that $\hskip 2.5pt\overline {\hskip -2.5pt A\hskip -.1pt}\hskip .1pt= A/J(A)$ is a separable field extension of ${\hskip 1.7pt\overline {\hskip -1.7pt R\hskip -.3pt}\hskip .3pt} = R/J(R)$. It follows from a recent result by Alkhamees and Singh that $A $ has a commutative $R$-subalgebra $R_{0}$ which is a chain ring such that $A = R_{0}+J(A)$ and $R_{0}\cap J(A) = J(R_{0}) = J(R)R_{0}$. The structure of $A$ in terms of a skew polynomial ring over $R_{0}$ is determined.