标题： 论近似代数的基础：公理、扩展与几何结构 显示英文标题

标题： On the Foundations of Approximate Algebra: Axioms, Extensions, and Geometric Structures

摘要： 在Inan [4]和Almahareeq-Peters-Vergili [1]最近工作的基础上，我们通过一个与代数相容的闭包算子$\Phi^{\!*}$来建立近似代数的严格公理基础，该算子满足(C1)-(C4a)以及平衡乘法公理(C4b)(仅对理想需要吸收性)。 我们的框架涵盖了近似模的理论及其同构定理，素谱上的近似Zariski拓扑的构造，以及局部化中的兼容局部化理论。 关键结果包括一个$\mathrm{T}_0$属性和谱的$\mathrm{T}_1$-准则，局部化中近似素理想的扩展-收缩双射，以及近似素根与幂零根的相等性。 该理论的实用性通过计算 $\mathrm{Spec}_{\!\Phi}(\mathbb{Z})$对于模闭包 $\Phi^{\!*}(A)=\langle A\rangle+m\mathbb{Z}$得到体现，这产生了一个有限离散空间——与经典 $\mathrm{Spec}(\mathbb{Z})$形成鲜明对比，后者是无限的且甚至不是 $\mathrm{T}_1$。 我们还概述了一条通向近似Nullstellensatz的路径，并记录了确保其评估-分离假设的闭包模型类。 显示英文摘要

摘要： Building on the recent works of Inan [4] and Almahareeq-Peters-Vergili [1], we develop a rigorous axiomatic foundation for approximate algebra via an algebra-compatible closure operator $\Phi^{\!*}$ satisfying (C1)-(C4a) together with the balanced multiplicativity axiom (C4b) (and absorption required only for ideals). Our framework encompasses a theory of approximate modules with their isomorphism theorems, the construction of an approximate Zariski topology on the prime spectrum, and a compatible theory of localization. Key results include a $\mathrm{T}_0$ property and a $\mathrm{T}_1$-criterion for the spectrum, an extension--contraction bijection for approximate prime ideals in localizations, and the equality of the approximate prime radical and the nilradical. The theory's utility is illustrated by computing $\mathrm{Spec}_{\!\Phi}(\mathbb{Z})$ for the modular closure $\Phi^{\!*}(A)=\langle A\rangle+m\mathbb{Z}$, which yields a finite discrete space -- in stark contrast to the classical $\mathrm{Spec}(\mathbb{Z})$, which is infinite and not even $\mathrm{T}_1$. We also outline a pathway toward an Approximate Nullstellensatz and record model classes of closures that ensure its evaluation--separation hypothesis.