There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes.
So, I read the solution to this problem: Prove that the only homomorphism between two cyclic groups with distinct prime orders is the trivial one
Would the argument to my question be basically the same thing since rings have an additive abelian group structure (although it is not necessarily cyclic) or would I need to expand (I'm going to guess its a yes). If so I'm thinking this should be done by contradiction, but I'm not entirely sure how to go about this or where the contradiction would come in. Any thoughts?

Hint: if $f$ is a homomorphism from a ring of characteristic $p$ into one of characteristic $q$, then $qf(1)=f(p1)=pf(1)=0$. Use this to show that $f(1)=0$.
