2025/09/21 by Phuc, Dang Vo
#08A05 #54E05 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13A15 #Rings and Algebras (math.RA) #Secondary 13C99
paper · doi:10.48550/arxiv.2509.25212
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 Φ * 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 T0 property and a T1-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 Spec Φ(ℤ) for the modular closure Φ *(A)=⟨ A⟩+mℤ, which yields a finite discrete space -- in stark contrast to the classical Spec(ℤ), which is infinite and not even T1. We also outline a pathway toward an Approximate Nullstellensatz and record model classes of closures that ensure its evaluation-separation hypothesis.