A ring homomorphism $f:R \to R'$ such that $R'$ has unity and zero divisors, what can we say about $f(1)$? 
I just finished the relatively simple problem above, and the proof of the final step comes down to $f(1)(1'-f(1))=0$ and we use the fact that $R'$ has no zero divisors to conclude $f(1)=1'$ where $1'$ is the unity in $R'$. Since, we had to use the fact that $R'$ has no zero divisors,

I was wondering if there are counterexamples if $R'$ does have zero divisors such that $f(1)$ is not the unity in $R'$.


Hint. $f:\mathbb Z_6\to\mathbb Z_6$ defined by $f(x)=3x$.
