Arithmetics in number systems with negative base

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set $\mathrm{F}\mathrm{i}\mathrm{n}(-\beta )$ and ${\mathbb{Z}}_{-\beta }$ of numbers having finite resp. integer $(-\beta )$-expansions. We show that $\mathrm{F}\mathrm{i}\mathrm{n}(-\beta )$ is trivial if $\beta$ is smaller than the golden ratio $\frac{1}{2}(1+\sqrt{5})$. For $\beta \ge \frac{1}{2}(1+\sqrt{5})$ we prove that ${\rm Fin}(-\beta)$isaring,onlyif$\beta$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-\beta)$isaringif$\beta$ is a quadratic Pisot number with positive conjugate. For quadratic Pisot units we determine the number of fractional digits that may appear when adding or multiplying two $(-\beta )$-integers.