Prime radicals of skew polynomial rings. (English) Zbl 1071.16024

Kwak, Tai Keun

Int. J. Math. Sci. 2, No. 2, 219-227 (2003).

Let \(\sigma\) be an endomorphism of a ring \(R\). \(R\) is called a \(\sigma(*)\)-ring if for \(a\in R\), \(a\sigma(a)\in P(R)\) implies \(a\in P(R)\). Here \(P(R)\) denotes the prime radical of \(R\). This is a generalization of the notion of a \(\sigma\)-rigid ring (\(a\sigma(a)=0\) implies \(a=0\)).
The author gives several characterizations of \(\sigma(*)\)-rings. Via several preliminary results, the main result is obtained which describes the prime radical of a skew polynomial ring \(R[x,\sigma]\): Let \(R\) be a \(\sigma(*)\)-ring with \(\sigma(P(R))\subseteq P(R)\). Then \(P(R[x,\sigma])=P(R)[x,\sigma]\) if and only if \(R[x,\sigma]\) is a 2-primal ring. An example is given to show the requirement that \(R\) be a \(\sigma(*)\)-ring is not superfluous.

Reviewer: Stefan Veldsman (Al-Khodh)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16W20	Automorphisms and endomorphisms
16N40	Nil and nilpotent radicals, sets, ideals, associative rings

Keywords:

prime radical; endomorphisms; rigid rings