Marshall Quotients of the Rings $\mathbb Z/n\mathbb Z$

Abstract

We study the Marshall quotient \[ M(n)=M(\mathbb Z/n\mathbb Z) \] obtained from the ring of integers modulo $n$ by quotienting by the square classes of non-zero-divisors. Using elementary arithmetic of square classes modulo prime powers and the Chinese Remainder Theorem, we give an explicit description of these quotients and classify several of their structural properties. We determine when the quotient relation is arithmetically elementary, when $M(n)$ is hyperbolic, when it can be real reduced, and when it can be formally real. We also analyze the subset of invertible classes together with zero, proving exactly when it is a submultiring, when it is a hyperfield, and when it is hyperbolic. The results provide a finite family of test examples for questions connecting multirings, special hyperfields, real semigroups, and abstract quadratic-form theory.