ON NIL-SEMICOMMUTATIVE RINGS

Abstract

Semicommutative and Armendariz rings are a generalization of reduced rings, and therefore, nilpotent elements play an important role in this class of rings. There are many examples of rings with nilpotent elements which are semicommutative or Armendariz. In fact, in [1], Anderson and Camillo prove that if R is a ring and n ≥ 2, then R[x]/(xn) is Armendariz if and only if R is reduced. In order to give a noncommutative generalization of the results of Anderson and Camillo, we introduce the notion of nilsemicommutative rings which is a generalization of semicommutative rings. If R is a nil-semicommutative ring, then we prove that niℓ(R[x]) = niℓ(R)[x]. It is also shown that nil-semicommutative rings are 2-primal, and when R is a nil-semicommutative ring, then the polynomial ring R[x] over R and the rings R[x]/(xn) are weak Armendariz, for each positive integer n, generalizing related results in [12].