| Abstrakt: |
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 D [ [ X ] ] consisting of all power series f = ∑ i = 0 ∞ a i X i in D [ [ X ] ] such that a i ∈ I for all large i. By definition, the polynomial ring D [ X ] and the power series ring D [ [ X ] ] are special cases of [ D ; I ] [ X ] when I = (0) and I = D, respectively. In this article, we investigate the ring R : = [ D ; I ] [ X ] 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 P [ [ X ] ] ∩ R = P R 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 D [ [ X ] ] , 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 0 = d ∈ D and S = { d n | n ≥ 0 }. 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. [ABSTRACT FROM AUTHOR] |
