## 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