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

Colloquium Mathematicae (2008)

• Volume: 113, Issue: 1, page 151-168
• ISSN: 0010-1354

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## Abstract

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A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $\left({\sum }_{i\ge 0}{a}_{i}{x}^{i}\right)\left({\sum }_{j\ge 0}{b}_{j}{x}^{j}\right)=0$ in R[x] (resp., in R[[x]]), then ${a}_{i}{b}_{j}=0$ 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 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.

## How to cite

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Tsiu-Kwen Lee, and Yiqiang Zhou. "A unified approach to the Armendariz property of polynomial rings and power series rings." Colloquium Mathematicae 113.1 (2008): 151-168. <http://eudml.org/doc/283978>.

@article{Tsiu2008,
abstract = {A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $(∑_\{i≥0\} a_i x^i)(∑_\{j≥0\} b_j x^j) = 0$ in R[x] (resp., in R[[x]]), then $a_i b_j = 0$ 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 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.},
author = {Tsiu-Kwen Lee, Yiqiang Zhou},
journal = {Colloquium Mathematicae},
keywords = {Armendariz rings; Gaussian rings; trivial extensions; skew polynomial rings; skew power series rings},
language = {eng},
number = {1},
pages = {151-168},
title = {A unified approach to the Armendariz property of polynomial rings and power series rings},
url = {http://eudml.org/doc/283978},
volume = {113},
year = {2008},
}

TY - JOUR
AU - Tsiu-Kwen Lee
AU - Yiqiang Zhou
TI - A unified approach to the Armendariz property of polynomial rings and power series rings
JO - Colloquium Mathematicae
PY - 2008
VL - 113
IS - 1
SP - 151
EP - 168
AB - A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $(∑_{i≥0} a_i x^i)(∑_{j≥0} b_j x^j) = 0$ in R[x] (resp., in R[[x]]), then $a_i b_j = 0$ 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 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.
LA - eng
KW - Armendariz rings; Gaussian rings; trivial extensions; skew polynomial rings; skew power series rings
UR - http://eudml.org/doc/283978
ER -

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