Note on the divisoriality of domains of the form k[[Xp,Xq]]k[[Xp,Xq]], k[Xp,Xq]k[Xp,Xq], k[[Xp,Xq,Xr]]k[[Xp,Xq,Xr]], and k[Xp,Xq,Xr]

Abstract

Let k be a field and X an indeterminate over k . In this note we prove that the domain k[[X p , Xq ]] (resp. k[X p , Xq ] ) where p, q are relatively prime positive integers is always divisorial but k[[X p , Xq , Xr ]] (resp. k[X p , Xq , Xr ] ) where p, q, r are positive integers is not. We also prove that k[[Xq , Xq+1, Xq+2]] (resp. k[Xq , Xq+1, Xq+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.