It is rather straightforward to prove that the only continuous Ring homomorphism from $\mathbb{R}$ onto itself is the identity, as its restriction to rational numbers is the identity.
I wonder if that still holds if one gives up on the continuity hypothesis? In other words, are all Ring endomorphisms on $\mathbb{R}$ continuous?
My intuition would be no, because Ring axioms seem to put loose enough constraint on transcendental numbers. Yet I haven't been able to find a construction for a non-continuous endomorphism.
Technical precision: the definition of Ring homomorphism meant in this question is 1-preserving.
