Turkish Journal of Mathematics
DOI
10.3906/mat-1503-15
Abstract
Let $k$ be a field and $X$ an indeterminate over $k$. In this note we prove that the domain $k[[X^{p}, X^{q}]]$ (resp. $k[X^{p}, X^{q}]$) where $p, q$ are relatively prime positive integers is always divisorial but $k[[X^{p}, X^{q}, X^{r}]]$ (resp. $k[X^{p}, X^{q}, X^{r}]$) where $p, q, r$ are positive integers is not. We also prove that $k[[X^{q}, X^{q+1}, X^{q+2}]]$ (resp. $k[X^{q}, X^{q+1}, X^{q+2}]$) is divisorial if and only if $q$ is even. These are very special cases of well-known results on semigroup rings, but our proofs are mainly concerned with the computation of the dual (equivalently the inverse) of the maximal ideal of the ring.
Keywords
Divisorial ideal, divisorial domain, Noetherian domain
First Page
38
Last Page
42
Recommended Citation
MIMOUNI, ABDESLAM
(2016)
"Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$,"
Turkish Journal of Mathematics: Vol. 40:
No.
1, Article 4.
https://doi.org/10.3906/mat-1503-15
Available at:
https://journals.tubitak.gov.tr/math/vol40/iss1/4