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 $k[X^{p}, X^{q}, X^{r}]$

Authors : Abdeslam MIMOUNI

Pages : 38-42

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $k$ be a field and $X$ an indeterminate over $k$. In this note we prove that the domain $k[[X^{p}, X^{q}]]$ insert ignore into journalissuearticles values(resp. $k[X^{p}, X^{q}]$); where $p, q$ are relatively prime positive integers is always divisorial but $k[[X^{p}, X^{q}, X^{r}]]$ insert ignore into journalissuearticles values(resp. $k[X^{p}, X^{q}, X^{r}]$); where $p, q, r$ are positive integers is not. We also prove that $k[[X^{q}, X^{q+1}, X^{q+2}]]$ insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(equivalently the inverse); of the maximal ideal of the ring.
Keywords : Divisorial ideal, divisorial domain, Noetherian domain