Geodesic
Locally Bounded Topologies on the Ring of Integers of a Global Field
Canadian journal of mathematics, Tome 33 (1981) no. 3, pp. 571-584

Voir la notice de l'article provenant de la source Cambridge University Press

A subset A of a topological ring R is bounded if given any neighborhood U of zero, there exists a neighborhood V of 0 such that AV ⊆ U and VA ⊆ U. The topology on R is locally bounded if there exists a bounded neighborhood of 0.We recall that a seminorm ‖··‖ on a ring R is a function from R into the non-negative real numbers satisfying ‖x‖ = 0 if x = 0, ‖–x‖ = ‖x‖, ‖x + y‖ ≦ ‖x‖ + ‖y‖ and ‖xy‖ ≦ ‖x‖ ‖y‖ for all x, y in R. A seminorm ‖··‖ on R is a norm on R if ‖x‖ = 0 implies x = 0. We note that a seminorm ‖··‖ on R defines a locally bounded topology on R, and a norm on R defines a Hausdorff, locally bounded topology on R. Two norms on R are said to be equivalent if they define the same topology on R. 
Cohen, Jo-Ann D. Locally Bounded Topologies on the Ring of Integers of a Global Field. Canadian journal of mathematics, Tome 33 (1981) no. 3, pp. 571-584. doi: 10.4153/CJM-1981-047-6
@article{10_4153_CJM_1981_047_6,
     author = {Cohen, Jo-Ann D.},
     title = {Locally {Bounded} {Topologies} on the {Ring} of {Integers} of a {Global} {Field}},
     journal = {Canadian journal of mathematics},
     pages = {571--584},
     year = {1981},
     volume = {33},
     number = {3},
     doi = {10.4153/CJM-1981-047-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-047-6/}
}
TY  - JOUR
AU  - Cohen, Jo-Ann D.
TI  - Locally Bounded Topologies on the Ring of Integers of a Global Field
JO  - Canadian journal of mathematics
PY  - 1981
SP  - 571
EP  - 584
VL  - 33
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-047-6/
DO  - 10.4153/CJM-1981-047-6
ID  - 10_4153_CJM_1981_047_6
ER  - 
%0 Journal Article
%A Cohen, Jo-Ann D.
%T Locally Bounded Topologies on the Ring of Integers of a Global Field
%J Canadian journal of mathematics
%D 1981
%P 571-584
%V 33
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-047-6/
%R 10.4153/CJM-1981-047-6
%F 10_4153_CJM_1981_047_6

[1] 1. Bourbaki, N., Algèbre commutative, Chapters 5 and 6 (Hermann, Paris, 1964). Google Scholar

[2] 2. Bourbaki, N., Algèbre commutative, Chapter 7 (Hermann, Paris, 1965). Google Scholar

[3] 3. Bourbaki, N., Topologie générale, Chapters 3 and 4 (Hermann, Paris, 1960). Google Scholar

[4] 4. Cohen, J., Locally bounded topologies on algebraic number fields and certain Dedekind domains, to appear. Google Scholar

[5] 5. Kowalsky, H. and Diirbaum, H., Arithmetische Kennzeichung von Korpertopologien, J. Reine Angew. Math. (1953), 135–152. Google Scholar

[6] 6. Mahler, K., Uber Pseudobewertungen, III, Acta. Math. 67 (1936), 283–328. Google Scholar

[7] 7. Nichols, E. and Cohen, J., Locally bounded topologies on global fields, Duke Math. J. U (1977), 853–862. Google Scholar

[8] 8. O'Meara, O. T., Introduction to quadratic forms (Springer-Verlag, Berlin, 1963). Google Scholar

[9] 9. P., Ribenboim, Algebraic numbers (Wiley-Interscience, New York, 1972). Google Scholar

[10] 10. Warner, S., Normability of certain topological rings, Proc. of the A.M.S. 33 (1972), 423–427. Google Scholar

Cité par Sources :