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

100%
Lee T.-K., Zhou Y.
Colloquium Mathematicum
|
2008
|
tom 113
|
nr 1
151-168
EN
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.
2
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On lifting of idempotents and semiregular endomorphism rings

100%
Lee T.-K., Zhou Y.
Colloquium Mathematicum
|
2011
|
tom 125
|
nr 1
99-113
EN
Starting with some observations on (strong) lifting of idempotents, we characterize a module whose endomorphism ring is semiregular with respect to the ideal of endomorphisms with small image. This is the dual of Yamagata's work [Colloq. Math. 113 (2008)] on a module whose endomorphism ring is semiregular with respect to the ideal of endomorphisms with large kernel.
3
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An intermediate ring between a polynomial ring and a power series ring

81%
Koşan M., Lee T.-K., Zhou Y.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 1
1-17
EN
Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = {$∑_{i≥0} r_i x^i ∈ R[[x]]$ : ∃ 0 ≤ n∈ ℤ such that $r_i∈ I$, ∀ i ≥ n}. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring property suggest a similar study to be carried out for the ring [R;I][x]. In this paper, we characterize when the ring [R;I][x] is semipotent, left Noetherian, left quasi-duo, principal left ideal, quasi-Baer, or left p.q.-Baer. New examples of these rings can be given by specializing to some particular ideals I, and some known results on polynomial rings and power series rings are corollaries of our formulations upon letting I = 0 or R.
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