ON NIL SKEW ARMENDARIZ RINGS

Abstract

Antoine [Nilpotent elements and Armendariz rings, J. Algebra319 (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. When the set of nilpotent elements of a ring R with an α-condition, namely α-compatibility, forms an ideal, we observe that R satisfies a nil Armendariz-type property, in the context of Ore extension R[x;α, δ]. For a 2-primal ring R with a derivation δ, R[x] is nil [Formula: see text]-skew Armendariz, and for a 2-primal ring R, R is nil α-skew Armendariz if and only if R[x] is nil [Formula: see text]-skew Armendariz, where α is an endomorphism of R with αk = id R, for some positive integer k. Moreover, we prove that a ring R is nil (α, δ)-skew Armendariz if and only if the n-by-n skew triangular matrix ring Tn(R, σ) is nil [Formula: see text]-skew Armendariz, for each endomorphism σ, with σ(1) = 1. A rich source of rings R, for which R[x] is nil [Formula: see text]-skew Armendariz, is provided.