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

Franz-Viktor Kuhlmann and Katarzyna Kuhlmann

Vol. 340 (2026), No. 2, 339–374

DOI: 10.2140/pjm.2026.340.339

Abstract

We investigate the 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 the 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 $I_1∕I_2$.

Keywords

ordered abelian group, cut, final segment, valuation ring, ideal, quotients of ideals over valuation rings, annihilator

Mathematical Subject Classification

Primary: 06F20, 13F30

Secondary: 12J25, 13A15

Milestones

Received: 9 April 2025

Revised: 6 October 2025

Accepted: 6 October 2025

Published: 26 November 2025