A group topology on a ring
is said to have ideal closure property in case the closure of an ideal is the
intersection of all maximal ideals containing it. Hinrichs considered such group
topologies on rings C(X) of continuous real-valued functions defined over
completely regular Hausdorff spaces. He gave a characterization of such
topologies with ideal neighbourhoods at zero in C(X), and showed that
there exists in C(X) a group topology with ideal closure property with the
largest collection of open ideals. His results are indeed true in a wider class of
rings—viz. semisimple commutative rings with unit element whose structure
spaces of maximal ideals (with hull-kernel topology) are Hausdorff. This
generalization is achieved by making use of a characterization by Gillman of such
rings.
