An Examination of Semiring Structures Derived from the Integers by Sequential Adjunction of Identity and Absorbing Elements

We conduct a systematic investigation of two algebraic structures constructed sequentially from the ring of integers Z. The first structure, S, is the globalization G(Z), obtained by adjoining an element T defined as the additive identity and multiplicative absorber. We establish through verification of the axioms that S is a commutative unital standard semiring. A comprehensive analysis of its algebraic properties demonstrates that S is a Principal Ideal Semiring (PIS), an integral semidomain, and zerosumfree, with all ideals being subtractive. Its Krull dimension is determined to be 2. We provide a complete classification of its congruences. We analyze its factorization properties, proving that S is not a Unique Factorization Domain (UFD); specifically, the element 0Z is shown to be prime but reducible, and it lacks a factorization into irreducibles. The ideal zeta function is computed as ζS (s) = ζ(s). The action of the unit group U (S) ∼= Z/2Z identifies the set of fixed points (singlets) A = {0Z, T }, which forms a sub-semiring isomorphic to the Boolean semiring B. The second structure, S′, is constructed by adjoining a universal absorbing element Ω to S (the absorber adjunction A(S)). We prove that S′ is a commutative unital hemiring, but not a standard semiring, as the additive identity (T ) is distinct from the multiplicative absorber (Ω). We analyze S′, proving it is a PIS and zerosumfree. We establish that no proper ideal in S′ is subtractive, a consequence of the presence of an additive absorber. Its Krull dimension is determined to be 3. The singlets of S′ form an idempotent sub-hemiring A′ = {0Z, T , Ω}, isomorphic to the extended Boolean semiring Bext. Generalizations to rings of integers OK in algebraic number fields are examined. We prove that the class semigroup of S(OK ) is isomorphic to the class group Cl(K). We conclude with a discussion of the topological and categorical implications, demonstrating how the sequential application of the functors G and A systematically increases the Krull dimension by introducing new generic points, satisfying Kdim(A(G(OK ))) = Kdim(OK ) + 2.