Formally real rings and their real closure
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'.] Section 3, `Uniqueness of real closures for some formally real rings', establishes for certain classes of rings that any two of the just mentioned \(\Gamma\) are isomorphic. The general uniqueness question remains open but it is shown for example that the real closures of products of formally real integral domains are isomorphic; similarly the closures \(D\) of a real subring \(A\) tightly sitting in the product of algebraically closed fields of characteristic 0. For the latter situation, in section 4, `Factorization of monic polynomials', it is shown that any monic polynomial \(f(T)\in D[T]\) can be written as a product of monic polynomials of degree \(\leq 2\). Section 5 applies the results, as do several other remarks, to rings of continuous functions. The paper seems carefully written and contains many illuminating examples and remarks; oddly, the well known article of \textit{T. Y. Lam} [Rocky Mt. J. Math. 14, 767-814 (1984; Zbl 0577.14016)] which contains in its sections 1, 2, 3 some of the results reproved and/or close to those of the paper, is not cited.
scientific article; zbMATH DE number 878118