Abstract
Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring consisting of all power series
in
such that
for all large i. By definition, the polynomial ring
and the power series ring
are special cases of
when
and I = D, respectively. In this article, we investigate the ring
in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that
is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring
the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring RS, where I = dD with
and
Closely related to R, the ring RS is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, RS has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise RS is a PID.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgment
The authors would like to thank the referee for his/her remarks, which help us improve the presentation of the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).