Formally real rings and their real closure
type: Item
instance of: scholarly article
title: Formally real rings and their real closure
zbMATH Open document ID: 0856.13003
DOI: 10.1007/BF00113916
author: K. Varadarajan
author: N. Sankaran
published in: Acta Mathematica Hungarica
publication date: 1997-02-24T00:00:00Z
review text: Recall that a ring \(A\) (all rings meant to be commutative with 1) is called (formally) real if \(\sum^k_1a^2_j=0\) \((a_j \in A)\) implies \(a_j=0\) for all \(j\) and that a field \(K\) is real closed if it is real and does not admit a proper real algebraic extension. It is well known that every real field can be imbedded in a real closed field \(\Lambda\) such that the extension of \(\Lambda/K\) is algebraic and that any two real closures of \(K\) are isomorphic. The authors propose to develop an analogous theory for a certain class of rings. A ring extension \(B \supseteq A\) is tight if any nonzero ideal of \(B\) has nonzero intersection with \(A\). The authors call a ring real closed if it is real but does not admit a proper integral tight real extension; a real closed \(\Gamma\) such that the extension \(\Gamma \supseteq A\) is tight and integral is called a real closure of \(A\). In section 1, `Generalities on tight extensions', one finds e.g. these results: An integral domain that is an integral extension is a tight extension; a tight extension of an integral domain is an integral domain; the intermediate rings of tight integral extensions of a reduced ring extend the latter tightly. A criterion for when the integral closure of a subring in a product of fields extends the ring tightly is also given. Section 2, `Formally real rings', proves that a real integral domain \(A\) is real closed iff it has a real closed quotient field containing \(A\) integrally closed. The authors now show part of their desired analog, modestly called lemma 5: Every real ring \(A\) has a real closure. -- The proof is by showing that any maximal formally real subring \(\Gamma \) of a total integral closure \(B\) of \(A\) serves as a real closure. [Totally integrally closed rings we introduced by \textit{E. Enochs}, Proc. Am. Math. Soc. 19, 701-706 (1968; Zbl 0157.08501) for the purpose of extending the notion of algebraic closure to rings. The present paper draws heavily on Enochs'…
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13J99
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 878118
zbMATH Keywords: formally real ring
zbMATH Keywords: tight ring extension
zbMATH Keywords: factorization of monic polynomials
zbMATH Keywords: real closure
zbMATH Keywords: formally real integral domains
zbMATH Keywords: rings of continuous functions
reviewed by: Alexander Kovačec
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00113916
OpenAlex ID: W2041809745
description: scientific article; zbMATH DE number 878118