Topological dense subsets of a ring

Abstract

Some properties of this topology are discussed. Amongst others, it is shown that proper closed ideals is more an exception than a rule. This topology is not compatible with separation: A ring is T₀ if and only if (a) = (b) implies a = b and T₁ if and only if it is discrete. The discrete rings are completely characterized and it is shown that a homomorphism is continuous if and only if the kernel is a closed ideal.