A representation theorem for Chain rings

Yousef Alkhamees ¹   ; Hanan Alolayan ²   ; Surjeet Singh ²

¹ Department of Mathematics King Saud University P.O. Box 2455, Riyadh 11451 Kingdom of Saudi Arabia
² Department of Mathematics King Saud University PO Box 2455, Riyadh 11451 Kingdom of Saudi Arabia

Colloquium Mathematicum, Tome 96 (2003) no. 1, pp. 103-119

Résumé

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.

Détail

DOI : 10.4064/cm96-1-10

Keywords: ring called chain ring local sided artinian principal ideal ring commutative chain ring faithful r algebra which chain ring hskip overline hskip hskip hskip separable field extension hskip overline hskip hskip hskip follows recent result alkhamees singh has commutative r subalgebra which chain ring cap structure terms skew polynomial ring determined