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

Volume 113 / 2008

Tsiu-Kwen Lee, Yiqiang Zhou Colloquium Mathematicum 113 (2008), 151-168 MSC: Primary 16S36, 16S99, 16U99. DOI: 10.4064/cm113-1-9


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.


  • Tsiu-Kwen LeeDepartment of Mathematics
    National Taiwan University
    Taipei 106, Taiwan
    Member of
    Mathematics Division (Taipei Office)
    National Center for Theoretical Sciences
  • Yiqiang ZhouDepartment of Mathematics and Statistics
    Memorial University of Newfoundland
    St. John's, NF, Canada A1C 5S7

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