Radicals in Ore extensions
Let \(R\) be a ring, \(\sigma \) an automorphism of \(R\) and \(\delta \) a \(\sigma \)-derivation of \(R.\) This means \(\delta \) is an additive map satisfying the product rule \(\delta (ab)=\sigma (a)\delta (b)+\delta (a)b\) for all \(a,b\in R\). By \(R[x;\sigma ,\delta ]\) is denoted the Ore extension of \(R\), subject to the left skewing condition \(xa=\sigma (a)x+\delta (a)\). If \(\delta =0,R[x;\sigma ]\) is called a skew polynomial ring. If \(\sigma=id_{R}\), \(R[x;\delta ]\) is called the differential polynomial ring. The radicals of these rings have been studied extensively and here the authors continue these investigations, in particular for the prime, Levitzki, strongly prime and uniformly strongly prime radicals of the Ore extension. A central problem in the radical theory of polynomial and related rings is also addressed here, namely to find the connections between the radical of the base ring \(R\) and the radical of the Ore extension.\N\NAt first, the new concepts \(\Omega \)-prime ideal and \(\Omega\)-prime radical of a ring \(R\) is defined to obtain connections between the prime radical of the Ore extension \(R[x;\sigma ,\delta \)] and the \(\Omega \)-prime radical of the base ring \(R\). Based on these results, the \(\Omega \)-\(LS\)-prime ideal, the \(\Omega \)-strongly prime ideal and the \(\Omega \)-uniformly strongly prime ideal of a ring \(R\) are defined to provide formulas for the \(LS\)-prime radical, the strongly prime radical and the uniformly strongly prime radical of the Ore extension. For polynomial and related rings, the so called Amitsur property is a significant and useful property. For the current context, a radical \(F\) is said to satisfy the \(\N\delta \)-Amitsur property if, for every ring \(R\) and for any derivation \(\delta \) of \(R\), \(F(R[x;\delta ])=(F(R[x;\delta ])\cap R)[x;\delta ].\) It is shown that the \(\Omega \)-(uniformly) strongly prime radical of \(R[x;\sigma,\delta ]\) has the Amitsur property. Most of the results are obtained under the requirement that \(\sigma \delta =\delta \sigma .\)
scientific article; zbMATH DE number 7910545
scientific article