Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images

V. K. Zakharov

Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 677-720.

Résumé

In this article we consider metaregular and countably-divisible extensions generated by a regular quotient ring of the ring of continuous functions in the spirit of Fine–Gillman–Lambek. The corresponding pre-images of maximal ideals are considered in connection with these extensions. These pre-images are called small absolutes and a-nonconnected coverings. To characterize these structures a new topological structure is introduced for Aleksandrov spaces with a precovering. In this connection we introduce the notion of a non-connected covering of step type. In the first part of the article we give a characterization of a small absolute as a relatively countably non-connected covering (Theorem 1). We also give a description of the absolute (Theorem 2) and of Aleksandrov pre-images of maximal ideals of Hausdorff–Sierpinski ring extensions (Theorem 3). In the second part we give a characterization of an $a$-non-connected pre-image as an absolutely countably non-connected covering (Theorem 4). Descriptions are also given of Baire and Borel pre-images generated by the classical Baire and Borel measurable extensions (Theorem 5).

Bibliographie

[1] Kuratovskii K., Topologiya, t. 1, Mir, M., 1966 | MR

[2] Jacobs K., Measure and integral, Academic Press, N.Y., 1978 | MR

[3] Semadeni Z., Banach spaces of continuous functions, Polish Sci. Publ., Warszawa, 1971 | MR | Zbl

[4] Burbaki N., Integrirovanie, Gl. III–V, IX, Nauka, M., 1977

[5] Arens R. F., “Operations induced in funcution classes”, Monatsh. Math., 55:1 (1951), 1–19 | DOI | MR | Zbl

[6] Zakharov V. K., “$cr$-obolochki koltsa nepreryvnykh funktsii”, DAN SSSR, 294:3 (1987), 531–534 | MR | Zbl

[7] Zakharov V. K., “Svyaz mezhdu polnym koltsom chastnykh koltsa nepreryvnykh funktsii, regulyarnym popolneniem i rasshireniem Khausdorfa–Serpinskogo”, UMN, 45:6 (1990), 133–134 | MR | Zbl

[8] Zakharov V. K., “Universalno-izmerimoe rasshirenie i rasshirenie Aresna banakhovoi algebry nepreryvnykh funktsii”, Funkts. analiz i ego prilozh., 24:2 (1990), 83–84 | MR | Zbl

[9] Zakharov V. K., “Svyazi mezhdu rasshireniyami Lebega i rasshireniyami Borelya pervogo klassa i mezhdu sootvetstvuyuschimi im proobrazami”, Izv. AN SSSR. Ser. matem., 54:5 (1990), 928–956 | MR | Zbl

[10] Zakharov V. K., Koltsa chastnykh i delimye obolochki koltsa nepreryvnykh funktsii, Avtoreferat dokt. dissert., Sankt-Peterburgskii gos. universitet, Sankt-Peterburg, 1991, 33 pp.

[11] Zakharov V. K., “Regulyarnoe i berovskoe rasshirenie koltsa nepreryvnykh funktsii kak koltsa chastnykh odnogo tipa”, UMN, 46:6 (1991), 209–210 | MR | Zbl

[12] Aleksandrov A. D., “Additive functions in abstract spaces. I—III”, Matem. sb., 8 (1940), 303–348 ; 9 (1941), 563–628 ; 13 (1943), 169–238 | MR | MR

[13] Fine N. J., Gillman L., Lambek J., Rings of quotients of rings of functions, McGill Univ. Press, Montreal, 1965 | MR

[14] Zaharov V. K., “On functions connected with absolute, Dedekind completion and divisible envelope”, Per. Math. Hung., 18:1 (1987), 17–26 | DOI | MR

[15] Ponomarev V. I., “Ob absolyute topologicheskogo prostranstva”, DAN SSSR, 153:5 (1963), 1013–1016 | MR | Zbl

[16] Zakharov V. K., Koldunov A. V., “Sekventsialnyi absolyut i ego kharakterizatsii”, DAN SSSR, 253:2 (1980), 280–284 | MR | Zbl

[17] Dashiell F., Hager A., Henriksen M., “Order–Cauchy completions of rings and vector lattices of continuous functions”, Can. J. Math., 32:3 (1980), 657–685 | MR | Zbl

[18] Zaharov V. K., “On functions connected with sequential absolute, Cantor completion and classical ring of quotients”, Per. Math. Hung., 19:2 (1988), 113–133 | DOI | MR

[19] Zakharov V. K., “Topologicheskie proobrazy, sootvetstvuyuschie klassicheskim rasshireniyam koltsa nepreryvnykh funktsii”, Vestn. Mosk. un-ta. Ser. 1, 1990, no. 1, 44–47 | MR | Zbl

[20] Zakharov V. K., “Proobraz Gordona prostranstva Aleksandrova kak okruzhaemoe nakrytie”, Izv. RAN. Ser. matem., 56:2 (1992), 427–448 | Zbl

[21] Zakharov V. K., “Proobrazy, svyazannye s polnym koltsom chastnykh, regulyarnym popolneniem i rasshireniyami Khausdorfa–Serpinskogo i Bera”, UMN, 48:5 (1933), 171–172

[22] Lambek I., Koltsa i moduli, Mir, M., 1971 | MR | Zbl

[23] Feis K., Algebra: koltsa, moduli i kategorii, Mir, M., 1977

[24] Veksler A. I., Zakharov V. K., “Regulyarnye popolneniya kommutativnykh polupervichnykh kolets”, Vestn. Mosk. un-ta. Ser. 1, 1981, no. 1, 98–99

[25] Zakharov V. K., “Funktsionalnoe predstavlenie regulyarnogo popolneniya modulei bez krucheniya Utumi”, Izv. VUZov. Matematika, 5 (1982), 22–29 | Zbl

[26] Zakharov V. K., “Schetno-delimoe rasshirenie i rasshirenie Bera koltsa i banakhovoi algebry nepreryvnykh funktsii kak delimaya obolochka”, Algebra i analiz, 5:6 (1993), 121–138 | MR

[27] Delfosse J.-P., “Caractérizations d'anneaux de fonctions continues”, Ann. Soc. Sci. Bruxelles. Ser. 1, 89:3 (1975), 364–368 | MR | Zbl

[28] Zakharov V. K., “Funktsionalnoe predstavlenie ravnomernogo popolneniya maksimalnogo i schetno-plotnogo modulei chastnykh modulya nepreryvnykh funktsii”, UMN, 35:4 (1980), 187–188 | MR | Zbl

[29] Hausdorff F., “Über halbstetige Functionen und deren Verallgemeinerung”, Math. Z., 4 (1915), 292–309 | MR

[30] Sierpin'ski W., “Sur les fonctions développables en séries absolument convergentes de fonctions continues”, Fund. Math., 2 (1921), 15–27 | MR | Zbl

[31] Zaharov V. K., “Alexandrovian cover and Sierpin'skian extension”, Studia Sci. Math. Hung., 24 (1989), 93–117 | MR | Zbl

[32] Zakharov V. K., Koldunov A. V., “Kharakterizatsiya $\sigma $-nakrytiya kompakta”, Sib. matem. zhurn., 23:6 (1982), 91–99 | MR

[33] Zaharov V. K., Koldunov A. V., “Characterization of the $\sigma $-cover of a compact”, Math. Nachr., 107 (1982), 7–16 | DOI | MR

[34] Gordon H., “The maximal ideal space of a ring of measurable functions”, Amer. J. Math., 88:4 (1966), 827–843 | DOI | MR | Zbl