# Characteristic of a ring: intuitive explanation

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.

• Your best bet might be to compute the characteristic of lots of examples to get a feel for it. – mathematician Jan 19 '15 at 14:21
• You can think about "walking in circles". You can know that you are walking in circles if you cross the starting point after a finite number of steps. – Integral Jan 19 '15 at 14:44
• Let $n>1$ be an integer and ${\mathbb Z}_n=\{0,1,\ldots, n-1\}$ equiped with multiplication and adition modulo $n$. Then this is an example of a ring with characteristic $n$. If $n$ is a prime, then it is actually a field. – Janko Bracic Jan 19 '15 at 15:11

Quotients of $\Bbb Z$ are a natural source of rings with different characteristics, of course.
The choice of $0$ to represent the case when there is no finite period is purely a conventional one. See also Why “characteristic zero” and not “infinite characteristic”?