ON NIL-SEMICOMMUTATIVE RINGS

Year 2012, Volume: 11 Issue: 11, 20 - 37, 01.06.2012

R. Mohammadi A. Moussavi M. Zahiri

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].

Keywords

nil semicommutative ring, semicommutative ring, weak Armendariz ring, weak α-rigid ring

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• R. Mohammadi, A. Moussavi, M. Zahiri Department of Pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University Tehran, Iran, P.O. Box 14115-134. e-mails: [e-mail] (R. Mohammadi) [e-mail], [e-mail] (A. Moussavi) [e-mail] (M. Zahiri)