[Math] Characteristics of ring homomorphisms

Question

Let R, R' be rings with unity and f : R → R' a ring homomorphism. Show that
Char(R') divides Char(R).

I dont get this questions, given that characteristics can only be prime or $0$ how do you prove one divides the other

Answer 1

If $R$ is an integral domain, then the characteristic can only be prime or zero. For arbitrary rings, the characteristic doesn't need to be. The characteristic is defined as the smallest integer $n$ for which $1+\dots+1=0$. For instance, $\Bbb Z/n\Bbb Z$ has characteristic $n$.

In other words, $\mathrm{char}(R)$ is the order of the element $1$ in the underlying group $(R,+)$ of $R$. Hence the positive characteristic case follows from the following more general fact about groups:

If $\phi:G\to H$ is a group homomorphism and $g\in G$ has finite order, then the order of $\phi(g)$ divides that of $g$.

and the zero characteristic case is immediate because all integers divide zero.