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

Tsiu-Kwen Lee ¹   ; Yiqiang Zhou ²

¹ Department of Mathematics National Taiwan University Taipei 106, Taiwan and Member of Mathematics Division (Taipei Office) National Center for Theoretical Sciences
² Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NF, Canada A1C 5S7

Colloquium Mathematicum, Tome 113 (2008) no. 1, pp. 151-168

Résumé

A ring $R$ 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.

Détail

DOI : 10.4064/cm113-1-9

Keywords: ring called armendariz resp armendariz power series type whenever sum sum resp paper deals unified generalization concepts see definition known results armendariz rings extended general situation results obtained consequences instance proved ring armendariz power series type injective endomorphism sigma ring proved sigma armendariz armendariz power series type sigma rigid sense krempa