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On \(\alpha\)-nilpotent elements and \(\alpha\)-Armendariz rings. (English) Zbl 1323.16018
A well-known result of Armendariz shows that a reduced ring \(R\) has the property that if \(f(x)g(x)=0\) for \(f(x),g(x)\in R[x]\), then \(ab=0\) for any coefficients \(a\) and \(b\) of \(f(x)\) and \(g(x)\), respectively. Rings with this latter property are now called Armendariz rings. Subsequently this notion, as well as the set of nilpotent elements in an Armendariz ring, have been generalized and extended in many ways; in particular also to skew polynomial rings.
In this paper the authors study and add to such generalizations. In particular they investigate \(\overline\alpha\)-nilpotent elements in a skew polynomial ring \(R[x;\alpha]\), where \(\overline\alpha\) is the monomorphism induced by the monomorphism \(\alpha\) of an Armendariz ring \(R\). Furthermore, for an endomorphism \(\beta\) of \(R\), the notion \(\beta\)-nil-Armendariz is defined as a natural generalization of a nil-Armendariz ring to a skew-polynomial ring \(R[x;\beta]\). Many examples are provided and the main line of investigation is to see how this notion transfers to related rings.
MSC:
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16U80 Generalizations of commutativity (associative rings and algebras)
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