Nilpotent elements and Armendariz rings

We study the structure of the set of nilpotent elements in Armendariz rings and introduce nil-Armendariz as a generalization. We also provide some new examples by proving that if D is a K-algebra and n ≥ 2, the coproduct D *K K 〈 x | xn = 0 〉 is Armendariz if and only if D is a domain with K {set minus} {0} as its group of units. Finally we study the conditions under which the polynomial ring over a nil-Armendariz ring is nil-Armendariz, which is related to a question of Amitsur. © 2008 Elsevier Inc. All rights reserved.