Vol. 24, No. 1, 1968

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Ideal neighbourhoods in a ring

H. Subramanian

Vol. 24 (1968), No. 1, 173–176

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.

Mathematical Subject Classification
Primary: 16.98
Received: 14 November 1966
Published: 1 January 1968
H. Subramanian