Semigroups from full lattices in commutative ${\mathbb Q}$-algebras

TL;DR

This paper explores the structure of semigroups formed by full lattices in finite dimensional commutative ${ m Q}$-algebras, extending classical results like the Jordan-Zassenhaus theorem to non-separable cases and applications to matrix conjugacy classes.

Contribution

It generalizes the Jordan-Zassenhaus theorem to non-separable algebras and analyzes the semigroup structure of full lattices in these algebras.

Findings

The semigroup of full lattices forms a commutative semigroup.

Extension of the Jordan-Zassenhaus theorem to non-separable algebras.

Application to conjugacy classes of integer matrices.

Abstract

The full lattices in a finite dimensional commutative ${\mathbb Q}$-algebra form a commutative semigroup. In the case of an algebraic number field the top part of a certain quotient semigroup is the class group. For a separable algebra some basic results, especially the Jordan-Zassenhaus theorem, are known for this quotient semigroup. This paper considers also algebras which are not separable. It studies the commutative semigroup of full lattices in such an algebra and also the quotient semigroup. This leads in this commutative, but not separable situation to a certain extension of the Jordan-Zassenhaus theorem. One application concerns $GL_n({\mathbb Z})$-conjugacy classes of regular integer $n\times n$ matrices.