Document Zbl 1121.16037

Power series rings satisfying a zero divisor property. (English) Zbl 1121.16037

Commun. Algebra 34, No. 6, 2205-2218 (2006).

E. P. Armendariz showed that a reduced ring \(R\) has the following property: If \(f(x)\) and \(g(x)\) are polynomials with coefficients in \(R\) and \(f(x)g(x)=0\), then the product of every coefficient of \(f(x)\) with every coefficient of \(g(x)\) is \(0\). Rings whose polynomials satisfy this property are now known as Armendariz rings. In this paper, the authors study rings \(R\) whose power series satisfy the corresponding property. For brevity we shall refer to such rings as being PSWA (power-serieswise Armendariz).
Clearly PSWA rings are Armendariz, and examples are given to show that the reverse implication is not true. It is shown that reduced rings are PSWA, and connections with other classes of rings are investigated. The final section gives some methods for constructing examples of PSWA rings.

Reviewer: A. W. Chatters (Bristol)

Cited in 38 Documents

MSC:

16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)
16S36	Ordinary and skew polynomial rings and semigroup rings
16D25	Ideals in associative algebras
16P60	Chain conditions on annihilators and summands: Goldie-type conditions

Keywords:

polynomial rings; power series rings; power-serieswise Armendariz rings; reduced rings; zero divisors; acc on annihilator ideals; minimal prime ideals; polynomial ring extensions

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