Arithmetic of cuts in ordered abelian groups and of ideals over valuation rings

TL;DR

This paper studies the algebraic structure of cuts in ordered abelian groups and ideals in valuation rings, providing conditions for solutions to certain equations and inequalities, and applying these to compute annihilators of quotient modules.

Contribution

It introduces new results on the existence, uniqueness, and maximality of solutions for equations involving cuts and ideals, linking ordered group theory with valuation ring ideal theory.

Findings

Conditions for solutions to equations involving cuts in ordered abelian groups.

Characterization of solutions for ideal equations in valuation rings.

Method to compute annihilators of quotient modules.

Abstract

We investigate existence, uniqueness and maximality of solutions $T$ for equations $S_1+T=S_2$ and inequalities $S_1+T\subseteq S_2$ where $S_1$ and $S_2$ are final segments of ordered abelian groups. Since cuts are determined by their upper cut sets, which are final segments, this gives information about the corresponding equalities and inequalities for cuts. We apply our results to investigate existence, uniqueness and maximality of solutions $J$ for equations $I_1 J=I_2$ and inequalities $I_1 J\subseteq I_2$ where $I_1$ and $I_2$ are ideals of valuation rings. This enables us to compute the annihilators of quotients of the form $I_1/I_2\,$.