# Ring homomorphisms map non units to non units

A ring homomorphism maps units to units.

I was wondering if it implies that it maps non units to non units. I tried to find a counter example because I think the answer should be no but couldn't find one. Any help is appreciated.

• Well, what if one ring is a field and the other isn't? Jun 15 '14 at 8:13
• What ring homomorphisms have you looked at? Off the top of my head, I think you have to try hard to find examples that aren't counterexamples: I think the only two classes that are "easy" to find are homomorphisms from a field, and the embedding from a ring into the polynomial ring over it.
– user14972
Jun 15 '14 at 8:19

An easy example is $f:\mathbb{Z} \to \mathbb{Q}$ with $f(m) := m$
Can you describe the ring morphisms $\mathbb C[x]\to\mathbb C$?
Can you describe the ring morphisms $\mathbb Z\to\mathbb Z/2\mathbb Z$?