Bélair, Luc Rings of \(p\)-adic functions. (Anneaux de fonctions \(p\)-adiques.) (French) Zbl 0868.03017 J. Symb. Log. 60, No. 2, 484-497 (1995). Summary: We study first-order properties of the quotient rings \({\mathcal C}(V)/{\mathcal P}\) by a prime ideal \(\mathcal P\), where \({\mathcal C}(V)\) is the ring of \(p\)-adic valued continuous definable functions on some affine \(p\)-adic variety \(V\). We show that they are integrally closed Henselian local rings, with a \(p\)-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy \(\forall x,y\) \((x|y^2\vee y|x^2)\). Cited in 1 ReviewCited in 4 Documents MSC: 03C60 Model-theoretic algebra 13J07 Analytical algebras and rings 13L05 Applications of logic to commutative algebra 12J10 Valued fields Keywords:rings of \(p\)-adic functions; first-order properties; quotient rings; prime ideal; \(p\)-adic valued continuous definable functions; affine \(p\)-adic variety; integrally closed Henselian local rings; \(p\)-adically closed residue field; field of fractions; valuation rings PDF BibTeX XML Cite \textit{L. Bélair}, J. Symb. Log. 60, No. 2, 484--497 (1995; Zbl 0868.03017) Full Text: DOI OpenURL References: [1] Géométrie algébrique réelle (1987) [2] On the L-adic spectrum 40 (1986) [3] Anneaux p-adiquement clos et anneaux de fonctions définissables 56 pp 539– (1991) [4] Les nombres p-adiques (1975) · Zbl 0313.12104 [5] On the structure of semialgebraic set over p-adic fields 53 pp 1138– (1988) [6] DOI: 10.1016/0168-0072(89)90061-4 · Zbl 0704.03017 [7] Algebra and order: proceedings of the first international symposium on ordered algebraic structures, Luminy-Marseille, 1984 pp 175– (1986) [8] Sheaves of continuous definable functions 53 pp 1165– (1988) [9] Rings of continuous functions (1960) [10] Methods in Mathematical logic, Proceedings, Caracas, 1983 1130 pp 76– (1985) [11] Jounal für Reine und Angewandte Mathematik 369 pp 154– (1986) [12] DOI: 10.1090/conm/008/653174 · Zbl 0485.14007 [13] DOI: 10.1007/BF01389133 · Zbl 0537.12011 [14] DOI: 10.1016/0168-0072(83)90019-2 · Zbl 0538.03028 [15] DOI: 10.1016/0022-4049(83)90058-0 · Zbl 0525.14015 [16] Partially ordered rings and semi-algebraic geometry (1979) · Zbl 0415.13015 [17] DOI: 10.1016/0168-0072(88)90043-7 · Zbl 0656.03023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.