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

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.