Uniqueness of real closure * of Baer regular rings

TL;DR

This paper investigates conditions under which the real closure * of Baer regular rings is unique, providing characterizations based on prime and real spectra, and exploring examples beyond real closed rings.

Contribution

It offers new characterizations for the uniqueness of real closure * in Baer regular rings using spectral properties, expanding understanding beyond real closed rings.

Findings

Characterizations of domains with unique real closure *

Conditions for Baer regular rings to have unique real closure *

An example showing regular rings can have unique real closure * without being f-rings

Abstract

It was pointed out in my last paper that there are rings whose real closure * are not unique. In [4] we also discussed some example of rings by which there is a unique real closure * (mainly the real closed rings). Now we want to determine more classes of rings by which real closure * is unique. The main results involve characterisations of domains and Baer regular rings having unique real closure *, and an example showing that regular rings need not be f-rings in order to have a unique real closure *. The main objective here is to find characterisation for uniqueness of real closure * for real regular rings that will primarily only require information of the prime spectrum and the real spectrum of the ring.