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/