ON NIL SKEW ARMENDARIZ RINGS

Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319 (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 alpha-condition, namely alpha-compatibility, forms an ideal, we observe that R satisfies a nil Armendariz-type property, in the context of Ore extension R[x; alpha, delta]. For a 2-primal ring R with a derivation delta, R[x] is nil delta-skew Armendariz, and for a 2-primal ring R, R is nil alpha-skew Armendariz if and only if R[x] is nil alpha-skew Armendariz, where a is an endomorphism of R with alpha(k) = id(R), for some positive integer k. Moreover, we prove that a ring R is nil (alpha, delta)-skew Armendariz if and only if the n-by-n skew triangular matrix ring T-n(R, sigma) is nil (alpha, delta)-skew Armendariz, for each endomorphism sigma, with sigma(1) = 1. A rich source of rings R, for which R[x] is nil (alpha, delta)-skew Armendariz, is provided.