Algebraic Geometry Over Hyperrings

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued’. This paper largely consisits of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme $(X,{\mathcal{O}}_X)$ and prove that $\mathrm{\Gamma }(X,{\mathcal{O}}_X)\simeq R$ for any integral affine hyperring scheme $X=SpecR$.