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

Claus Hertling, Khadija Larabi

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.

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

Abstract

The full lattices in a finite dimensional commutative -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 -conjugacy classes of regular integer matrices.
Paper Structure (10 sections, 32 theorems, 203 equations, 2 figures)

This paper contains 10 sections, 32 theorems, 203 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Full lattices in finite dimensional ${\mathbb Q}$-vector spaces
  3. Finite dimensional commutative ${\mathbb Q}$-algebras
  4. Commutative semigroups
  5. Full lattices and orders in ${\mathbb Q}$-algebras, some semigroups
  6. An extension of the Jordan-Zassenhaus theorem
  7. Localization
  8. Exact sequences for an order and a smaller order
  9. Dividing out the radical
  10. Sufficiently high powers of full lattices are invertible

Key Result

Lemma 1.2

(a) The idempotents in ${\mathcal{L}}(A)$ are the orders. The idempotents in ${\mathcal{E}}(A)$ are the $\varepsilon$-classes of the orders. (b) $L$ is invertible in ${\mathcal{L}}(A)$ if and only if $[L]_\varepsilon$ is invertible in ${\mathcal{E}}(A)$. Then $e_L={\mathcal{O}}(L)$ and $e_{[L]_\vare

Figures (2)

  • Figure 3.1: Radical filtration in Example \ref{['t3.4']}
  • Figure 3.2: Socle filtration in Example \ref{['t3.4']}

Theorems & Definitions (39)

  • Lemma 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Lemma 3.3
  • Example 3.4
  • ...and 29 more