## Abstract

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.

$S$ 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.

Original language | English |
---|---|

Pages (from-to) | 17-31 |

Number of pages | 15 |

Journal | Algebras and Representation Theory |

Volume | 19 |

Publication status | Published - 2016 |

## Keywords

- prime ideal
- semigroup algebra
- nilpotent group
- classical Krull dimension