Jacobson Density Theorem - Topological Characterization

Topological Characterization

A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem. There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring of End(_DU) by identifying each element of R with the D linear transformation it induces by right multiplication. If U is given the discrete topology, and if UU is given the product topology, and End(_DU) is viewed as a subspace of UU and is given the subspace topology, then R acts densely on U if and only if R is dense set in End(_DU) with this topology.