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.

  • $\begingroup$ Your best bet might be to compute the characteristic of lots of examples to get a feel for it. $\endgroup$ – mathematician Jan 19 '15 at 14:21
  • $\begingroup$ 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. $\endgroup$ – Integral Jan 19 '15 at 14:44
  • $\begingroup$ 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. $\endgroup$ – Janko Bracic Jan 19 '15 at 15:11

It is simply a statement about the maximum additive order of something in the ring.

Quotients of $\Bbb Z$ are a natural source of rings with different characteristics, of course.

The most physical analogy that comes to mind is modular arithmetic. If you're familiar with any sort of cyclic behavior that repeats after finitely many steps, you can view the characteristic of the ring as a "period" of the cyclic behavior.

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”?

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