If G is a Hausdorff topological group, saying that G is monothetic is equivalent to saying there exists a homomorphism $f: \mathbb{Z} \to G$ with dense image.
A multiplication can be naturally defined in $f(\mathbb{Z})$ by $f(a) \cdot f(b) := f(ab)$. This forces f to act like a ring homomorphism, therefore turning $f(\mathbb{Z})$ into a ring.
With that in mind, is there any sufficient condition that allows this multiplication to be extended through density to the whole of G? And when is it guaranteed that this extension turns G into a topological ring? I don't mind assuming G to be locally compact in case that helps.