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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?

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  • $\begingroup$ Can I ask you to describe how $\mathcal{F}$ acts on morphisms? $\endgroup$
    – Todd Trimble
    Jan 18, 2014 at 1:13
  • $\begingroup$ I apologize. my functor does not have a reasonable action on morphism, so I revise my question. $\endgroup$ Jan 18, 2014 at 1:23
  • $\begingroup$ Consider a simple $R$ (field, finite factor, ...) or a product of such. You can at most distinguish the number of factors, but not the differences among simple rings. So you have a "commutative" concept not very useful in non-commutative geometry. $\endgroup$
    – user46855
    Mar 1, 2014 at 4:42

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