Ring homomorphisms map non units to non units

Question

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.

Answer 1

Can you describe the ring morphisms $\mathbb C[x]\to\mathbb C$?

Can you describe the ring morphisms $\mathbb Z\to\mathbb Z/2\mathbb Z$?

Answer 2

Every integral domain is embedded in a 'field of fractions' of its elements, so any such non-field will have a natural (injective) homomorphism to a field which fails this property.

An easy example is $f:\mathbb{Z} \to \mathbb{Q}$ with $f(m) := m$