# 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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topTsiu-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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