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

M. Tamer Koşan¹ ; Tsiu-Kwen Lee² ; Yiqiang Zhou³

¹ Department of Mathematics Gebze Institute of Technology Gebze, Kocaeli, Turkey
² Department of Mathematics National Taiwan University Taipei 106, Taiwan Member of Mathematics Division (Taipei Office) National Center for Theoretical Sciences
³ Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NL A1C 5S7, Canada

Colloquium Mathematicum, Tome 130 (2013) no. 1, pp. 1-17

Résumé

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 $R[[x]]$: $$[R;I][x]:=\Big\{\sum_{i\ge 0}r_ix^i\in R[[x]]: \exists 0\le n\in {\mathbb Z}\ \text {such that}\ r_i\in I,\, \forall i\ge n\Big\}.$$ 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 $R$.

Détail

DOI : 10.4064/cm130-1-1

Keywords: respectively denote ring polynomials ring power series indeterminate ring ideal denote following subring sum exists mathbb text forall polynomial power series rings extreme cases where there ideals neither nor isomorphic results characterizing polynomial rings power series rings certain ring property suggest similar study carried out ring paper characterize ring semipotent noetherian quasi duo principal ideal quasi baer baer examples these rings given specializing particular ideals known results polynomial rings power series rings corollaries formulations letting