An intermediate ring between a polynomial ring and a power series ring
Volume 130 / 2013
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 $R[[x]]$: $$[R;I][x]:=\Big\{\sum_{i\ge 0}r_ix^i\in R[[x]]: \exists 0\le n\in {\mathbb Z}\ \text {such that}\ r_i\in I,\, \forall i\ge n\Big\}.$$ 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 $R$.
