Some examples of reduced rings. (English) Zbl 0859.16019

Krempa, Jan

Algebra Colloq. 3, No. 4, 289-300 (1996).

The paper is concerned with a study of necessary and sufficient conditions for a ring \(R\) to be reduced, that is, to have no nonzero nilpotents. The emphasis is on certain important classes of constructions: skew polynomial rings, certain graded rings and crossed products, group rings and semigroup rings. A sample result reads as follows. Let \(S=R[x,\sigma,\delta]\) be a skew polynomial ring over a ring \(R\), where \(\sigma\) is a monomorphism of \(R\) and \(\delta\) is a \(\sigma\)-derivation of \(R\). Then \(S\) is reduced if and only if \(R\) is reduced and \(\sigma\) is rigid, that is, \(rr^\sigma\neq 0\) for every \(0\neq r\in R\). Moreover, in this case every minimal prime ideal of \(S\) is of the form \(P[x,\sigma,\delta]\) for a minimal prime \(P\) in \(R\).

Reviewer: J.Okniński (Warszawa)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
20M25	Semigroup rings, multiplicative semigroups of rings
16S35	Twisted and skew group rings, crossed products
16N40	Nil and nilpotent radicals, sets, ideals, associative rings

Keywords:

skew polynomial rings; graded rings; crossed products; group rings; semigroup rings; minimal prime ideals