Nilpotent elements and Armendariz rings.
type: Item
instance of: scholarly article
title: Nilpotent elements and Armendariz rings.
zbMATH Open document ID: 1157.16007
author: Ramon Antoine
published in: Journal of Algebra
publication date: 2008-07-11T00:00:00Z
review text: Let \(R\) denote an associative ring with \(1\), and let \(\text{nil}(R)\) denote the set of nilpotent elements. Further, let \(f(x)=\sum_{i=0}^ma_ix^i,g(x)=\sum_{j=0}^nb_jx^j\in R[x]\) denote two arbitrary polynomials. One says that \(R\) is an Armendariz ring if \(f(x)g(x)=0\) implies that \(a_ib_j=0\) for all \(i\) and \(j\). Similarly, one says that \(R\) is a weak Armendariz ring if \(f(x)g(x)=0\) implies \(a_ib_j\in\text{nil}(R)\) for all \(i\) and \(j\). The author makes the following definition, which is a mixture of these two: a ring \(R\) is nil-Armendariz if whenever \(f(x)g(x)\in\text{nil}(R)[x]\) then \(a_ib_j\in\text{nil}(R)\) for all \(i\) and \(j\). The author proves that Armendariz rings are nil-Armendariz, gives examples to show the converse implication does not hold, classifies when certain coproducts of algebras over fields are Armendariz, and proves that \(\text{nil}(R)\) is a subring when \(R\) is a (nil-)Armendariz ring. The author also proves that if \(R\) is nil-Armendariz then \(R/\text{Nil}^*(R)\) is Armendariz, but \(R/\text{Nil}_*(R)\) is not in general Armendariz. A number of interesting and clarifying examples are provided.
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 5300800
zbMATH Keywords: Armendariz rings
zbMATH Keywords: nil-Armendariz rings
zbMATH Keywords: polynomial rings
zbMATH Keywords: nilpotent elements
reviewed by: Pace P. Nielsen
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.01.019
OpenAlex ID: W1965857771
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DOI: 10.1016/J.JALGEBRA.2008.01.019
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description: scientific article; zbMATH DE number 5300800