Proving that a zero ring cannot have a homomorphism to a unital ring.

Question

Q: Prove that a zero ring cannot have a homomorphism to a unital ring.

I don't know how to prove this. If $f:R\to S$ is the homomorphism, and $f(0_R)=0_S$, then I don't see which homomorphism property is not being satisfied.

Any help would be much appreciated.

Answer 1

The usual definition of a homomorphism of unital rings requires that $f(1)=1$, so if you regard the zero ring as a unital ring (with $0=1$) then your map doesn't satisfy this condition.

Some people (though I think this is a bit old-fashioned) require that $0\neq1$ in a unital ring (the only effect of this is to rule out the zero ring as a unital ring).