A unified approach to the Armendariz property of polynomial rings and power series rings
Tom 113 / 2008
Colloquium Mathematicum 113 (2008), 151-168
MSC: Primary 16S36, 16S99, 16U99.
DOI: 10.4064/cm113-1-9
Streszczenie
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.
