Geodesic
Connection between the classical ring of quotients of the ring of continuous functions and Riemann integrable functions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 161-176.

Voir la notice de l'article provenant de la source Math-Net.Ru

The small Fine–Gillman–Lambek extension generated by the classical ring of quotients, and the Riemann extension generated by Riemann $\mu$-integrable functions are both characterized as divisible envelopes of the same type of the ring of all bounded continuous functions on the Aleksandrov space. This shows the similarity of these extensions that are rather different by their origin. 
@article{FPM_1995_1_1_a7,
     author = {V. K. Zakharov},
     title = {Connection between the classical ring of quotients of the ring of continuous functions and {Riemann} integrable functions},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {161--176},
     publisher = {mathdoc},
     volume = {1},
     number = {1},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a7/}
}
TY  - JOUR
AU  - V. K. Zakharov
TI  - Connection between the classical ring of quotients of the ring of continuous functions and Riemann integrable functions
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 1995
SP  - 161
EP  - 176
VL  - 1
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a7/
LA  - ru
ID  - FPM_1995_1_1_a7
ER  - 
%0 Journal Article
%A V. K. Zakharov
%T Connection between the classical ring of quotients of the ring of continuous functions and Riemann integrable functions
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 1995
%P 161-176
%V 1
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a7/
%G ru
%F FPM_1995_1_1_a7
V. K. Zakharov. Connection between the classical ring of quotients of the ring of continuous functions and Riemann integrable functions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 161-176. http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a7/

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

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

[3] Burbaki N., Integrirovanie, gl. III–V, IX, Nauka, M., 1977

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

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

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

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

[8] 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 | DOI | MR | Zbl

[9] 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

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

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

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

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

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

[15] Zakharov V. K., “Rasshirenie Arensa koltsa nepreryvnykh funktsii”, Algebra i analiz, 4:1 (1992), 135–153 | MR | Zbl

[16] Zakharov V. K., Koltsa chastnykh i delimye obolochki koltsa nepreryvnykh funktsii, Dis. ... dokt. fiz.-mat. nauk, S.-P., 1991, 210 pp. | Zbl

[17] 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 | Zbl

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

[19] Zakharov V. K., “Topologicheskie proobrazy, sootvetstvuyuschie klassicheskim rasshireniyam koltsa nepreryvnykh funktsii”, Vestnik Mosk. un-ta, ser. 1, 1990, no. 1, 44–45 | MR | Zbl

[20] 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

[21] Zakharov V. K., “Delimost na schetno-plotnye idealy i schetnaya ortopolnota modulei”, Matem. zametki, 30:4 (1981), 481–496 | MR | Zbl

[22] Zakharov V. K., “Funktsionalnaya kharakterizatsiya absolyuta, vektornye reshetki funktsii so svoistvom Bera i kvazinormalnykh funktsii i moduli chastnykh nepreryvnykh funktsii”, Tr. Mosk. mat. ob-va, 45 (1982), 68–104 | MR

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