TITLE: Characteristic of a ring: intuitive explanation
QUESTION [6 upvotes]: I know the following definition of characteristic of a ring: it is the smallest positive $n$ such that $$\underbrace{a+\cdots+a}_{n \text{ summands}} = 0$$
for every element a of the ring, if $n$ exists, otherwise it is zero. However, I don't understand what is the "intuitive" meaning of this. Could you give a physical analogy of anything that may help to see what it is.

REPLY [3 votes]: I agree with the above answer completely and would like to add little bit more to it.
Most familiar rings to us are $\mathbb{Z}, \mathbb{Q}$ and $\mathbb{Z}/n\mathbb{Z}$ and once derived from them (function rings, polynomial rings, matrix rings etc). It's not difficult to see that all of these are somehow connected to (or obtain from) $\mathbb{Z}.$ We can make this comparison mathematically precise by observing that for any ring $R$ (with unity) there is a canonical ring homomorphism $\varphi : \mathbb{Z}\to R.$
The kernel of this map (the data that we lost during the transformation) is an ideal of (the principal ideal domain) $\mathbb{Z}.$ Therefore we have a smallest (non-negative) integer that generates it, say $m$. This is exactly the characteristic of $R.$
Moreover, by the first isomorphism theorem we have that $$\mathbb{Z}/m\mathbb{Z}\cong \operatorname{im}\varphi,$$ which is a sub-ring of $R.$  Also since $$m.1_R=0$$ the ring $R$ become a $\mathbb{Z}/m\mathbb{Z}$-algebra. An instructive example for this is Boolean rings. Furthermore, if $R$ is an integral domain then $m$ must be a prime number or zero. So, this number $m$ (characteristic) carries some important information about the ring $R.$