I am currently self-learning ring theory and I stumbled over two equivalent statements describing the characteristic of a ring R.
Given a commutative ring R, the characteristic of R is the smallest positive integer m such that $ma = 0$ for any element $a$ in R.
Given a commutative ring R, the characteristic of R is the smallest positive integer m such that $f(m) = 0$, where $f$ is the unique homomorphism from $Z$ to $R$.
How can we show that these two statements are true? I have already shown that ker($f$)=$mZ$ (i.e. all integer n such that $f(n) = 0$ are multiples of m). I know that this step will be important in the proof, but how do I continue? I am currently stuck on this step? Are there some properties of $f$ that I should be using?
Thank you!
I guess you're just missing the connection between $ma$ in item 1 and $f(m)$ in item 2.
Strictly speaking, "$ma$" means $a+a+\cdots$ where $m$ copies of $a$ appear. On the other hand, since $R$ (apparently) has identity, and we know that $f(1)$ is that identity, and also that $f(1)m=m$ for all $m\in M$. It quickly follows from here that $ma$ (in the sense of item 1) equals $f(m)a$, so it is easy to "confuse" the $\mathbb Z$ module action on $M$ with the $f(\mathbb Z)$ module action on $M$.
They are not identical though: the kernel of the $\mathbb Z$ action can be nonzero (if the characteristic is positive) but the action of $f(\mathbb Z)$ is faithful. The link between the two is that
$$ \text{kernel of the $\mathbb Z$ action}=\ker f $$
In either case, they both refer to the ideal $m\mathbb Z$ of $\mathbb Z$, and that is the role $m$ plays.
(Incidentally, most definitions of characteristic also append the convention "... and if no such integer exists, the characteristic is $0$.")