Arithmetic of cuts in ordered abelian groups and of ideals over valuation rings
Franz-Viktor Kuhlmann, Katarzyna Kuhlmann
TL;DR
The article develops a unified framework linking cuts and final segments in ordered abelian groups to ideals in valuation rings, enabling a translation of equations $S_1+T=S_2$ and inclusions into the valuation-theoretic language via the value group $vK$ and $vI$. It introduces invariance groups ${\mathcal G}(M)$, the closure operations $\widehat{S}$ and $T^{\diamondsuit}$, and analyzes solvability and maximality of the equations, even in higher rank settings, using deep closures and quotient reductions. For valuation-theoretic applications, it characterizes $I_1J=I_2$ and $I_1J\subseteq I_2$ through overrings $\mathcal{O}(I)$ and invariance data $vI$, yielding explicit formulas for annihilators $I_2:I_1$ and the associated closures $J^{\diamondsuit}$. The paper further shows how to compute annihilators of quotients $I_1/I_2$, with concrete criteria in principal versus nonprincipal cases, and applies these results to special cases arising in pr1 and pr2, including relationships to Kähler differentials and valuation extensions. Overall, the work provides practical, order-theoretic tools to analyze and compute annihilators and ideal relations in valuation-theoretic contexts.
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 $I_1/I_2\,$.