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]

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.