Let A be a commutative unital Fréchet algebra, i.e. a completely metrizable topological algebra. Our main result states that all ideals in A are closed if and only if A is a noetherian algebra
@article{bwmeta1.element.bwnjournal-article-smv138i3p293bwm, author = {W. \.Zelazko}, title = {A characterization of commutative Fr\'echet algebras with all ideals closed}, journal = {Studia Mathematica}, volume = {141}, year = {2000}, pages = {293-300}, zbl = {0984.46032}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv138i3p293bwm} }
Żelazko, W. A characterization of commutative Fréchet algebras with all ideals closed. Studia Mathematica, Tome 141 (2000) pp. 293-300. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv138i3p293bwm/
[00000] [1] M. Akkar et C. Nacir, Continuité automatique dans les limites inductives localement convexes de Q-algèbres de Fréchet, Ann. Sci. Math. Québec 19 (1995), 115-130. | Zbl 0840.46031
[00001] [2] S. Banach, Théorie des Opérations Linéaires, Warszawa, 1932. | Zbl 0005.20901
[00002] [3] G. Carboni and A. Larotonda, An example of a Fréchet algebra which is a principal ideal domain, this issue, 265-275. | Zbl 0969.46041
[00003] [4] A. V. Ferreira and G. Tomassini, Finiteness properties of topological algebras, Ann. Scuola Norm. Sup. Pisa 5 (1978), 471-488. | Zbl 0397.46045
[00004] [5] H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer, 1971.
[00005] [6] A. Mallios, Topological Algebras. Selected Topics, North-Holland, 1986.
[00006] [7] E. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952). | Zbl 0047.35502
[00007] [8] S. Rolewicz, Metric Linear Spaces, PWN, 1972.
[00008] [9] H. H. Schaefer, Topological Vector Spaces, Springer, 1971.
[00009] [10] W. Żelazko, Selected Topics in Topological Algebras, Aarhus Univ. Lecture Notes 31, 1971. | Zbl 0221.46041
[00010] [11] W. Żelazko, On maximal ideals in commutative m-convex algebras, Studia Math. 58 (1976), 291-298. | Zbl 0344.46103
[00011] [12] W. Żelazko, On topologization of countably generated algebras, ibid. 112 (1994), 83-88. | Zbl 0832.46042
[00012] [13] W. Żelazko, On m-convexity of commutative real Waelbroeck algebras, Comm. Math., submitted. | Zbl 1001.46035
[00013] [14] W. Żelazko, Characterizations of Q-algebras of type F and of F-algebras with all ideals closed, Acta Comm. Univ. Tartuensis, to appear. | Zbl 1044.46041