An ordered semiring is a commutative semiring equipped with a compatible preorder. The notion of ordered semiring generalises those of distributive lattice and of commutative ring, and enables us to unify certain aspects of lattice theory and ring theory. The ideals of an ordered semiring A form an integral commutative quantale Idl(A), and similarly, the radical ideals of A form a (spatial) frame Rad(A). We characterise Idl and Rad as the left adjoints of the forgetful functors from the categories of integral commutative quantales and of frames to that of ordered semirings and subadditive morphisms between them. The (sober) topological space pt(Rad(A)) = Spec(A) corresponding to Rad(A) is a space of prime ideals of A.