We prove that a real or complex unital F-algebra is a Q-algebra if and only if all its maximal one-sided ideals are closed.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-1-6, author = {W. \.Zelazko}, title = {A characterization of Q-algebras of type F}, journal = {Studia Mathematica}, volume = {162}, year = {2004}, pages = {73-79}, zbl = {1064.46030}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-1-6} }
W. Żelazko. A characterization of Q-algebras of type F. Studia Mathematica, Tome 162 (2004) pp. 73-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-1-6/