On nil skew Armendariz rings. (Q2920382)

Let \(R\) be a ring with \(1\), \(\alpha\) an endomorphism, \(\delta\) an \(\alpha\)-derivation of \(R\), and \(R[x;\alpha,\delta]\) the Ore extension of \(R\). Then \(R\) is called nil \((\alpha,\delta)\)-skew Armendariz if for \(f(x)=\sum_{i=0}^ma_ix^i\) and \(g(x)=\sum_{j=0}^nb_jx^j\in R[x;\alpha,\delta]\), \(f(x)g(x)\in\text{nil}(R)[x;\alpha,\delta]\) implies \(a_ix^ib_jx^j\in\text{nil}(R)[x;\alpha,\delta]\) for each \(i\) and \(j\) where \(\text{nil}(R)\) is the set of nilpotent elements of \(R\). In particular, a nil \((id_R,0)\)-skew Armendariz ring is a nil-Armendariz ring. The authors show some conditions for \(R\) under which \(R\) is nil \((\alpha,\delta)\)-skew Armendariz.NEWLINENEWLINE Theorem 1. Let \(R\) be \(\alpha\)-compatible (i.e., for \(a,b\in R\), \(ab=0\) if and only if \(a\alpha(b)=0\)) and \(\text{nil}(R)\) is an ideal, then \(R\) is nil \((\alpha,\delta)\)-skew Armendariz.NEWLINENEWLINE Theorem 2. If \(R\) is \(\alpha\)-compatible and \(\alpha\)-skew Armendariz, then \(R\) is nil \((\alpha,0)\)-skew Armendariz.NEWLINENEWLINE Theorem 3. Let \(R\) be \((\alpha,\delta)\)-compatible (i.e., \(R\) is both \(\alpha\)- and \(\delta\)-compatible as defined similarly above) and \(\text{nil}(R)\) is a subring. Then, \(R\) is nil \((\alpha,\delta)\)-skew Armendariz if and only if \(R/\text{Nil}^*(R)\) is \((\overline\alpha,\overline\delta)\)-skew Armendariz where \(\text{Nil}^*(R)\) is the upper nil-radical of \(R\), and \(\overline\alpha\) and \(\overline\delta\) are induced by \(\alpha\) and \(\delta\), respectively.NEWLINENEWLINE Moreover, a nil \((\alpha,\delta)\)-skew Armendariz ring \(R\) is characterized in terms of \(R[x]\) and some subrings of the skew triangular matrix ring \(T_n(R,\sigma)\) where \(\sigma\) is an endomorphism of \(R\), and for all \(i\leq j\), \((a_{ij}),(b_{ij})\in T_n(R,\sigma)\), \((a_{ij})(b_{ij})=(c_{ij})\), such that \(c_{ij}=\sum_{k=0}^{j-i}a_{i(i+k)}\sigma^kb_{(i+k)j}\).