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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.

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