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On power serieswise Armendariz rings. (English) Zbl 1402.16017

In this paper, all rings are associative with identity elements. Recall that a ring \(R\) is power serieswise Armendariz ring if every \(\displaystyle f(X)=\sum_{i=0}^{\infty} a_iX^i\) and \(\displaystyle g(X)=\sum_{i=0}^{\infty} b_iX^i\in R[[X]]\) such that \(f(X)g(X)=0\), then \(a_ib_j=0\) for every \(i\) and \(j\). Let \(A\) be a ring and \(E\) be a bi-module. The idealization of \(E\) over \(A\) is the set \(A(+)E=\{(a,e);a\in A, e\in E\}\). It is a ring for pairwise addition and multiplication given by \((a,e)(b,f)=(ab,af+be)\). In this paper, the authors investigate the transfer property of power serieswise Armendariz to idealization, direct product of rings and homomorphic image. Their results generate new and original examples of power serieswise Armendariz rings.

MSC:

16S70 Extensions of associative rings by ideals
16U99 Conditions on elements
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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