We construct an example as announced in the title. We also indicate all right, left and two-sided ideals in this example.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm104-2-8, author = {W. \.Zelazko}, title = {An m-convex B0-algebra with all left but not all right ideals closed}, journal = {Colloquium Mathematicae}, volume = {106}, year = {2006}, pages = {317-324}, zbl = {1096.46027}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm104-2-8} }
W. Żelazko. An m-convex B₀-algebra with all left but not all right ideals closed. Colloquium Mathematicae, Tome 106 (2006) pp. 317-324. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm104-2-8/