Fields with continuous local elementary properties. I (Q1910279)

Notations and results from the author's [\(RC^*\)-fields, Algebra Logic 33, No. 4, 205-215 (1994); translation from Algebra Logika 33, No. 4, 367-386 (1994; Zbl 0827.12007)]\ are used in the paper under review. The majority of the results proved in the paper were announced in his paper [Model-theoretic properties of \(RC^*\)-fields, Dokl. Ross. Akad. Nauk 335, No. 2, 138-141 (1994; Zbl 0840.12005)]. In the first section of the article technical results concerning 1-embeddings of systems of the form \(\langle F, R, J\rangle\), where \(R\) is a subring of a field \(F\) and \(J\) is an ideal of \(R\), are collected. Some definitions and results concerning Boolean products are recalled in the second section. The third section is the core of the article. In this section theorems are proved which provide necessary and sufficient conditions for the extensions \(\langle F, R_W, J(R_W) \rangle\leq \langle F', R_{W'}, J(R_{W'}) \rangle\) to be 1-extensions or elementary extensions for the pairs \(\langle F, R_W \rangle, \langle F', R_{W'} \rangle\in RC^*\) under some natural conditions. Those families from \(RC^*\) are described and investigated whose local elementary properties are continuous.