On skew Hurwitz serieswise Armendariz rings. (Q2925272)

Let \(R\) be an associative ring with unity and \(\alpha\) an endomorphism of \(R\). The ring of skew Hurwitz series over \(R\), written as \((HR,\alpha)\), is the set of all functions \(f\colon\mathbb N\to R\) with respect to componentwise addition and multiplication \(fg\) given by NEWLINE\[NEWLINE(fg)(n)=\sum_{k=0}^n{n\choose k}f(k)\alpha^k(g(n-k))NEWLINE\]NEWLINE for all \(n\in\mathbb N\), \(\mathbb N\) is the set of non-negative integers. The subring of all the functions with finite support is denoted by \((hR,\alpha)\).NEWLINENEWLINE Analogous to the concept of an Armendariz ring, the authors define and study skew Hurwitz serieswise Armendariz rings; the latter being commutative rings \(R\) for which \(fg=0\) if and only if \(f(n)g(m)=0\) for all \(n,m\) where \(f,g\in HR\). For such a ring \(R\), certain radicals of \(HR\) and \(hR\) are determined and it is shown that these two rings fulfill the Köthe conjecture (no nonzero nil ideals implies no nonzero one-sided nil ideals). The transfer of many properties, for example like being symmetric, reversible, prime and Baer, between \(R\), \(HR\) and \(hR\) is also discussed.