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

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 $I_1\mathrm{/}{I}_2$.