We prove that a real or complex F-algebra has all left and right ideals closed if and only if it is noetherian.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-4, author = {W. \.Zelazko}, title = {A characterization of F-algebras with all one-sided ideals closed}, journal = {Studia Mathematica}, volume = {166}, year = {2005}, pages = {135-145}, zbl = {1074.46031}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-4} }
W. Żelazko. A characterization of F-algebras with all one-sided ideals closed. Studia Mathematica, Tome 166 (2005) pp. 135-145. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-4/