On weakened \((\alpha ,\delta)\)-skew Armendariz rings. (English) Zbl 1513.16029

Summary: In this note, for a ring endomorphism \(\alpha\) and an \(\alpha\)-derivation \(\delta\) of a ring \(R\), the notion of weakened \((\alpha,\delta)\)-skew Armendariz rings is introduced as a generalization of \(\alpha\)-rigid rings and weak Armendariz rings. It is proved that \(R\) is a weakened \((\alpha,\delta)\)-skew Armendariz ring if and only if \(T_n(R)\) is weakened \((\bar{\alpha},\bar{\delta})\)-skew Armendariz if and only if \(R[x]/(x^n)\) is weakened \((\bar{\alpha},\bar{\delta})\)-skew Armendariz ring for any positive integer \(n\).


16S36 Ordinary and skew polynomial rings and semigroup rings
16S50 Endomorphism rings; matrix rings
16S99 Associative rings and algebras arising under various constructions
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