Transparency of Polynomial Ring Over a Commutative Noetherian Ring

Abstract

In this paper, we discuss a stronger type of primary decomposition (known as transparency) in noncommutative set up. One of the class of noncommutative rings are the skew polynomial rings. We show that certain skew polynomial rings satisfy this type of primary decomposition. Recall that a right Noetherian ring $R$ is said to be \textit{transparent ring} if there exist irreducible ideals $I_{j}$, 1 $\leq j \leq n$ such that $\cap_{j = 1}^{n}I_{j} = 0$ and each $R/I_{j}$ has a right artinian quotient ring. Let $R$ be a commutative Noetherian ring, which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers). Let $\sigma$ be an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. Then we show that the skew polynomial ring $R[x;\sigma,\delta]$ is a transparent ring.