I have a question about rings (I assume ring without the requirement of multiplicative identity element).
Suppose, I want to show, for two rings, there is NO ring isomorphism (the rings are not isomorphic).
Let´s take an arbitrary homomorphism $h$. I think every ring isomorphism has to map zero to zero. So is it sufficient to show that there exist two elements $a,b$ such that $f(a + b) = 0$, but $f(a) + f(b)$ cannot be zero?
(The "+" on the left and right side are the additive operations of the two corresponding ring, they dont have to be defined the same btw).
Is my thinking correct?
Thank you.