环ℤ/nℤ的Marshall商
Marshall Quotients of the Rings $\mathbb Z/n\mathbb Z$
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中文总结 AI 辅助
该文研究模n整数环的Marshall商，给出其显式描述并分类结构性质，分析含零的可逆类子集，为相关代数问题提供有限测试例族。
中文摘要 AI 辅助
我们研究模n整数环的Marshall商M(n)=M(ℤ/nℤ)，该商通过对非零因子的平方类取商得到。利用素数幂模平方类的初等算术与中国剩余定理，我们给出这些商的显式描述并分类其若干结构性质。我们确定商关系何时为算术初等的、M(n)何时为双曲的、何时可实约化、何时形式实。我们还分析含零的可逆类子集，证明其何时为子多环、何时为超域、何时为双曲的。这些结果为关联多环、特殊超域、实半群与二次型抽象理论的问题提供了有限的测试例族。
英文摘要
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.
