A unified approach to the Armendariz property of polynomial rings and power series rings
Volume 113 / 2008
Abstract
A ring is called Armendariz (resp., Armendariz of power series type) if, whenever (\sum_{i\ge 0}a_ix^i)( \sum _{j\ge 0}b_jx^j)=0 in R[x] (resp., in R[[x]]), then a_ib_j=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 \sigma of a ring R and for n\ge 2, it is proved that R[x;\sigma ]/(x^n) is Armendariz iff it is Armendariz of power series type iff \sigma is rigid in the sense of Krempa.