# Are there discontinuous Ring endomorphisms on real numbers? [duplicate]

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.

• This is probably a better dupe target. – Arnaud D. Jan 22 at 18:55
• No, but there are discontinuous Abelian group homomorphisms from the additive group of $\Bbb R$ to itself. – Berci Jan 22 at 19:40