##
**Some examples of reduced rings.**
*(English)*
Zbl 0859.16019

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 |