Restriction of topological ring isomorphism

Question

If $\theta: R\to S$ is an isomorphism of topological rings then do we obtain a topological group isomorphism $\theta|_{R^{\times}}:R^{\times}\to S^{\times}$ by restricting to their groups of units?

I'd be very grateful if someone could clarify this for me.

Answer 1

Yes. $R\mapsto R^\times$ (with the obvious effect on morphisms) is functorial.

Less categorically, it is the nature of isomorphisms that they make the isomorphic objects indistinguishable and hence doing the same (suficiently abstract) thing to both gives isomorphic stuff again