On skew polynomial rings

TL;DR

This paper investigates the structure of prime ideals in skew polynomial rings over noetherian rings, establishing conditions under which linked prime ideals are extended from the base ring.

Contribution

It proves that in certain skew polynomial rings, linked prime ideals that are extended from the base ring must themselves be extended, clarifying ideal linkage behavior.

Findings

Linked prime ideals extended from R are themselves extended in S(R).

The main theorem applies to noetherian, link k-symmetric rings with automorphisms.

Provides a structural understanding of prime ideal linkage in skew polynomial rings.

Abstract

In this note we consider the links of prime ideals of certain skew polynomial rings and prove our main theorem, namely theorem [5], which states the following.Let R be a noetherian ring that is link k-symmetric and let {\sigma} be an automorphism of R.Let S(R) denote the skew polynomial ring R[x,{\sigma}]. Let B be a prime ideal of S(R) that is extended from R. Then, for a prime ideal D of S(R),there is a link D\rightarrowB in the ring S(R) implies that D is an extended prime ideal of S(R).