Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase:
For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$.
We can reapet the same Idea for a $C^{*}$ algebra $A$, instead of a ring R. In this case we require that the ideals are closed. Putting $A=C(X)$, we can show that this topology is the $C^{*}$ algebraic analogy of compact open topology, defined on the space of homeomorphisms of $X$.
My questions:
1 Under what conditions on a ring R, the operation of "inversion", as a map on AUT(R), is continuous?
2 To what extent, this topology is useful, as a ring theoretical invariant?