Real Closures of Semilocal Rings, and Extension of Real Places
نویسنده
چکیده
(A) All rings in this announcement are commutative and with 1. For any ring K we denote by W(K) the Witt ring of nondegenerate symmetric bilinear forms over K. DEFINITION 1. A signature o of K is a ring homomorphism from W(K) to Z. REMARK 1. If K is a field, the signatures correspond uniquely with the orderings of K [3], [9]. Thus Theorem 1 below generalizes the main results of Artin-Schreier's theory of ordered fields [1], We consider pairs (K, L of rings is a (connected) covering, if a is the inductive limit of finite etale connected extensions of K, as studied in Galois theory. We say that a homomorphism a :(K, a) -» (L, T) is a covering, if K -» L is a covering. DEFINITION 2. A rea/ closure of a pair (K, cr) is a covering a:(K, tr) -• (jR, p) such that (R, p) does not admit any coverings except isomorphisms. By Zorn's lemma any pair (K, a) has at least one real closure.
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