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A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever P i≥0 aix i P j≥0 bjx j = 0 in R[x] (resp., in R[[x]]), aibj = 0 for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism σ of a ring R and for n ≥ 2, it is proved that R[x; σ]/(xn) is Armendariz iff it is Armendariz of power series type iff σ is rigid in the sense of Krempa. Throughout, unless otherwise stated all rings are associative with unity and modules are unitary. The ring of polynomials (resp., power series) in indeterminate x over a ring R is denoted by R[x] (resp., R[[x]]). For an endomorphism σ of a ring R, R[x; σ] and R[[x; σ]] denote the (left) skew polynomial ring and (left) skew power series ring, in which the multiplication is subject to the condition that xr = σ(r)x for all r ∈ R. Following [15] (resp., [14]), a ring R is called Armendariz (resp., Armendariz of power series type) if, whenever ( ∑ i≥0 aix )( ∑ j≥0 bjx ) = 0 in R[x] (resp., in R[[x]]), aibj = 0 for all i and j. An Armendariz ring of power series type is also called a power-serieswise Armendariz ring in [11]. The two notions have been widely studied. This paper deals with a unified generalization of these rings: For an ideal I of a ring R, the notion of an I-Armendariz ring R is defined such that R is Armendariz iff R is 0-Armendariz and that R is Armendariz of power series Date: December 18, 2007. 1991 Mathematics Subject Classification. Primary 16S36, 16S99, 16U99.

A unified approach to the Armendariz property of polynomial rings and power series rings