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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.

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  • $\begingroup$ Your best bet might be to compute the characteristic of lots of examples to get a feel for it. $\endgroup$ Jan 19, 2015 at 14:21
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    $\begingroup$ 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, 2015 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. $\endgroup$ Jan 19, 2015 at 15:11
  • $\begingroup$ @Janko Bracic: I do not think the last claim in your comment is true. For a simple counter example consider $\mathbb{Z}_2\times\mathbb{Z}_2, $ which is not even an integral domain. But it's converse is true for fields of non-zero characteristic (even for such integral domains). $\endgroup$
    – Bumblebee
    Nov 29, 2020 at 18:29
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    $\begingroup$ @Bumblebee My claim is about $Z_n$, where $n\geq 2$ is a prime number. It is a very very well known fact. $\endgroup$ Nov 30, 2020 at 9:06

2 Answers 2

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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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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 $R.$

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  • $\begingroup$ I only see this concept during the first course of algebra. Later, people usually only refer to $\mathbb Q$-algebras as "of char 0" — invertibility is important. $\endgroup$
    – Yai0Phah
    May 13 at 19:40

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