Central Algebraic Geometry and Seminormality

TL;DR

This paper develops the theory of central ideals and introduces the concept of central seminormalization in commutative rings, connecting algebraic structures with regulous functions on real algebraic varieties.

Contribution

It introduces the notion of central seminormalization, provides a construction method via elementary central gluings, and establishes its existence in affine cases and real schemes.

Findings

Central seminormalization relates to regulous functions on real algebraic varieties.

A decomposition theorem for constructing central seminormalization is provided.

Existence of central seminormalization is proven for affine rings and real schemes.

Abstract

We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties. We provide a construction of the central seminormalization by a decomposition theorem in elementary central gluings. The existence of a central seminormalization is established in the affine case and for real schemes.