# An intermediate ring between a polynomial ring and a power series ring

M. Tamer Koşan; Tsiu-Kwen Lee; Yiqiang Zhou

Colloquium Mathematicae (2013)

- Volume: 130, Issue: 1, page 1-17
- ISSN: 0010-1354

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topM. Tamer Koşan, Tsiu-Kwen Lee, and Yiqiang Zhou. "An intermediate ring between a polynomial ring and a power series ring." Colloquium Mathematicae 130.1 (2013): 1-17. <http://eudml.org/doc/283949>.

@article{M2013,

abstract = {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.},

author = {M. Tamer Koşan, Tsiu-Kwen Lee, Yiqiang Zhou},

journal = {Colloquium Mathematicae},

keywords = {polynomial rings; power series rings; principal ideal rings; quasi-duo rings; semipotent rings; Noetherian rings; Noetherian modules; power series modules; quasi-Baer rings; quasi-Baer modules; Laurent polynomial rings; Laurent series rings},

language = {eng},

number = {1},

pages = {1-17},

title = {An intermediate ring between a polynomial ring and a power series ring},

url = {http://eudml.org/doc/283949},

volume = {130},

year = {2013},

}

TY - JOUR

AU - M. Tamer Koşan

AU - Tsiu-Kwen Lee

AU - Yiqiang Zhou

TI - An intermediate ring between a polynomial ring and a power series ring

JO - Colloquium Mathematicae

PY - 2013

VL - 130

IS - 1

SP - 1

EP - 17

AB - 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.

LA - eng

KW - polynomial rings; power series rings; principal ideal rings; quasi-duo rings; semipotent rings; Noetherian rings; Noetherian modules; power series modules; quasi-Baer rings; quasi-Baer modules; Laurent polynomial rings; Laurent series rings

UR - http://eudml.org/doc/283949

ER -

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