Arithmetics in number systems with a negative base

We study the numeration system with a negative base, introduced by Ito and Sadahiro. We focus on arithmetic operations in the sets and Z−β of numbers having finite resp. integer (−β)-expansions. We show that is trivial if β is smaller than the golden ratio . For we prove that is a ring, only if β is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that is a ring if β 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 (−β)-integers.