DECOMPOSABILITY OF EXTENSION RINGS

Abstract

Skew polynomial rings have invited attention of mathematicians and various properties of these rings have been discussed. The nature of ideals (in particular prime ideals, minimal prime ideals, associated prime ideals), primary decomposition and Krull dimension have been investigated in certain cases.

This article concerns transparent (decomposable) rings. Recall that a ring $\mathrm{R}$ is said to be a Transparent ring if in $\mathrm{R}$ there exist irreducible ideals $I_j,1\le j\le n$ such that ${\cap }_{j=1}^n{I}_j=0$ and each $R/I_j$ has a right Artinian quotient ring.

Now let $\mathrm{R}$ be a ring, which is an order in an Artinian ring $\mathrm{S}$. Let $\sigma$ and $\tau$ be automorphisms of $\mathrm{R}$ and $\delta$ be a $(\sigma ,\tau )$-derivation of $\mathrm{R}$; i.e. $\delta :R\to R$ is an additive mapping satisfying $\delta \left(ab\right)=\sigma \left(a\right)\delta \left(b\right)+\delta \left(a\right)\tau \left(b\right)$ for all $\text{a},b\in R$. We define an extension of $\mathrm{R}$, namely $R[x,\sigma ,\tau ,\delta ]=\{f=\sum _{i=0}^n{x}^i{a}_i,a_i\in R\}$, subject to the relation $ax=x\sigma \left(\tau \right(a\left)\right)+\delta \left(a\right)$ for all $a\in R$.

We show that if $\mathrm{R}$ is a commutative Noetherian $\mathrm{\mathbb{Q}}$-algebra, $\sigma$ and $\tau$ as usual, then there exists an integer $m\ge 1$ such that the extension ring $R[x,\alpha ,\beta ,\vartheta ]$ is a Transparent ring, where $\alpha ={\sigma }^m$, $\beta ={\tau }^m$ and $\vartheta$ is an $(\alpha ,\beta )$-derivation of $\mathrm{R}$ with $\alpha \left(\vartheta \right(a\left)\right)=\vartheta \left(\alpha \right(a\left)\right)$, and $\beta \left(\vartheta \right(a\left)\right)=\vartheta \left(\beta \right(a\left)\right)$, for all $a\in R$.