A note on nil power serieswise Armendariz rings

Abstract

A ring R is called nil power serieswise Armendariz if $\forall f = \sum\limits_{i = 0}^\infty {a_i X^i }$ and $g = \sum\limits_{i = 0}^\infty {b_i X^i }$ in R[[X]] such that f g ∈ Nil(R)[[X]], then a _i b _j ∈ Nil(R) for all i and j. In this note we characterize completely nil power serieswise Armendariz rings with their nilradical Nil(R) (where the nilradical is the set of nilpotent elements). We prove that a ring is nil power serieswise Armendariz if and only if Nil(R) is an ideal of R. We prove that each power serieswise Armendariz ring is nil power serieswise Armendariz and we give examples of nil power serieswise Armendariz rings.