MHB Understanding Rings: Example of Homomorphism/Isomorphism

[corkycorey101]

Can you give an example of two rings R and S, and a function f:R⟶S such that f(ab)=f(a)f(b) for all a,b ∈ R, but f(a+b)≠f(a)+f(b) for some a,b ∈ R. I know that it has to do with proving homomorphisms/isomorphisms but am confused how to come up with the actual example.

 

[Deveno]

Let $R = S =\Bbb Z$, the ring of integers, and set:

$f(k) = k^2$.

 

[corkycorey101]

Oh wow, that was easier than I thought. I was way over thinking it. Thank you so much!