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

Knowing that both are either prime or zero doesn't change what it means for one to divide the other; all of the same proof methods you know still apply. All this fact means is that you have additional methods of proof, since if $x$ and $y$ are nonnegative integers that are both either prime or zero, then $x | y$ if and only if $x=y$ or $y=0$.

However, characteristics can be composite too. For example, $\mathbb{Z} / 6 \mathbb{Z}$ (i.e. the ring of integers modulo 6) has characteristic 6.

The fact you have in mind is about fields, or more generally, about integral domains.

Answer 2

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.