Prime ideals in algebras determined by submonoids of nilpotent groups

The prime spectrum of the semigroup algebra $K[S]$ of a submonoid
$S$ of a finitely generated nilpotent group is studied via the
spectra of the monoid $S$ and of the group algebra $K[G]$ of the
group $G$ of fractions of $S$. It is shown that the classical
Krull dimension of $K[S]$ is equal to the Hirsch length of the
group $G$ provided that $G$ is nilpotent of class two. This uses
the fact that prime ideals of $S$ are completely prime. An
infinite family of prime ideals of a submonoid of a free nilpotent
group of class three with two generators which are not completely
prime is constructed. They lead to prime ideals of the
corresponding algebra. Prime ideals of the monoid of all upper
triangular $n\times n$ matrices with non-negative integer entries
are described and it follows that they are completely prime and
finite in number.