Nilpotent Elements and Skew Polynomial Rings

Abstract

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.