An Algebraic Structure Incorporating a Z/1Z-Symmetric Element Adjoined to the Integers: Construction, Analysis, and Generalizations

We present a systematic construction of an algebraic structure S obtained by adjoining an element T to the ring of integers Z. The binary operations on S are specified such that T functions simultaneously as the additive identity of the structure S (that is, x+T = x for all x ∈ S) and as the multiplicative absorbing element (that is, x × T = T for all x ∈ S). We establish through complete verification of all axioms that (S, +, ×) constitutes a commutative unital semiring. A detailed analysis of this semiring’s properties is provided, including the classification of its idempotents, the determination of its group of units, and the characterization of its ideal structure. We prove that S is a Principal Ideal Semiring and characterize its spectrum of prime ideals, determining its Krull dimension to be 2. Furthermore, we examine the canonical action of the group of units U (Z) ∼= Z/2Z extended to an action by automorphisms on the additive monoid of S. We prove that the set of fixed points (singlets) under this action, denoted by A, consists precisely of the integer zero 0Z and the adjoined element T . This characterizes T as an element exhibiting Z/1Z symmetry distinct from 0Z within this algebraic context. We conduct a complete analysis of the algebraic structure of A inherited from S, proving it is isomorphic to the Boolean semiring B. Connections to the theory of the field with one element F1 are examined, analyzing S as an object in the category of F1-algebras (commutative monoids with an absorbing element) and investigating its base extension to the category of rings, which is shown to be the monoid ring Z[(Z, ×)]. Topological aspects, including the analysis of the Zariski topology on Spec(S), and generalizations to arbitrary commutative rings R are examined, establishing that dim(SR) = dim(R) + 1 and determining the automorphism group Aut(SR) ∼= Aut(R).