Approximate Algebra via Closure Operators: An Axiomatic Theory of Modules and Geometry

Abstract

Building on the work of \.{I}nan and of Almahariq--Peters--Vergili, we develop an axiomatic framework for approximate algebra based on an algebra-compatible closure operator $\Phi^{\!*}$ on a unital ring. The operator is assumed to be extensive, monotone, idempotent, compatible with addition, and balanced with respect to left and right multiplication, while absorption is imposed only in the definition of approximate ideals. The first structural result is that the closure of every approximate ideal is an ordinary two-sided ideal, so that the approximate quotient $R/\!I$ is canonically the ordinary quotient $R/\Phi^{\!*}(I)$. The decisive prime-theoretic result is that every approximate prime ideal in a unital ring is automatically $\Phi^{\!*}$-closed. Consequently, for a commutative ring with unity, $\mathrm{Spec}_{\!\Phi}(R)=\{P\in\mathrm{Spec}(R):\Phi^{\!*}(P)=P\},$ and the approximate Zariski topology is exactly the subspace topology induced from the classical spectrum on this fixed-prime locus. We also develop a compatible module theory, distinguish carefully between classical quotients and approximate quotients, and prove a first isomorphism theorem for the induced homomorphism modulo $\Phi^{\!*}_{M'}(0)$, together with an approximate quotient version under a closed-kernel hypothesis. For ideal-translation closures $\Phi^{\!*}(A)=A+J$, the spectrum is $V(J)\cong\mathrm{Spec}(R/J)$; in particular, the modular closure $A\mapsto A+m\mathbb Z$ yields the finite discrete space of prime divisors of $m$. Finally, over an algebraically closed field, we prove $\mathrm{rad}_{\!\Phi}(I)=\sqrt{\Phi^{\!*}(I)}$, show that the evaluation--separation implication is automatic from the classical Hilbert Nullstellensatz, and establish that point-ideal closedness is equivalent to the exact identity $\mathrm{rad}_{\!\Phi}(I)=\mathrm I(V(I))$ for all ideals.