A Note on the Globalization Semiring G(Z)

Let S = ℤ ∪ {τ}, where τ is declared to be the additive identity and multiplicative absorber. We prove, in a self-contained manner, the classification of ideals and prime ideals of S, the description of its congruences and quotient semirings, the equality Frac(S) ≅ 𝔹, the determination of its local stalks, the infinite-dimensionality of the prime spectrum of its multiplicative monoid, the direct-product decomposition of the contracted monoid algebra, and the extension of the entire construction to rings of integers 𝒪_K of number fields, including the identification Cl(S(𝒪_K)) ≅ Cl(K) and the equality of zeta functions ζ_{S(𝒪_K)} = ζ_K. The proofs are complete; the references serve only to locate the surrounding literature on semirings, F₁-geometry, blueprints, and algebraic number theory.