I have a personal vendetta against the definition of 'characteristic' as stated above. First I'll state my (proposed) alternate definition, then I'll give some reasons for my dislike.
The Characteristic of a ring with identity
Given a ring R with identity e, let n be the additive order of e. By definition of additive order, n is a positive integer, or positive infinity. Let this n be known as the characteristic of R. (Part of the reason for this definition is that, for a given r ∈ R, n*r = n*r*e = r*(n*e) = 0R, so n is (a multiple of) the additive order of each element of R.)
My reasons for this definition include the following:
* There are no rings with identity which can also have characteristic 1. If we allow a ring to have characteristic 0 we have created a gap in the set of possible characteristics.
* 0 is not a positive integer. This also conflicts with how we define characteristic. (On the other hand, I admit that infinity is not a positive integer... at least it's positive though.)
* Using infinity for a characteristic matches up more closely with the usual idea behind the order of an element.
These are my ideas and reasons. Do not confuse them with those of the "professionals". Also, please glance at my Odd-Even Theorem writeup for more information regarding why I choose this definition of characteristic. As an alternative (if my ideas cause too much grief/stir), I might decide to use 'focus' as my term for the number described above. It makes no exceptional difference to me.
In unrelated news, we have Moore's theorem on integral domains:
"Let R be an integral domain and n be its characteristic (as defined in this writeup). Then for all positive integers t, either n divides t or for all a, b in R, t*a=t*b implies a=b."
Let R be an integral domain and n be its characteristic (as defined in this writeup). Let t be a positive integer. If n divides t then we are done, except we note that in this case, there exist a and b in R such that t*a=t*b(=0), but a != b ("!=" is "does not equal"). Now we consider the case where n does not divide t. Suppose that there exist a and b in R such that t*a=t*b but a != b. Then t*a-t*b=0R=t*(a-b), which means that t is a potential candidate for the characteristic of R. Let c=a-b. n does not divide t, so that means that d=gcd(n,t)<n. From number theory, we know that there exist integers x and y such that n*x+t*y=d. We already know that t*c=t*(a-b)=0R. We further note that n*1R*c=n*c=0R. Now we note that d*c=(n*x+t*y)*c=0R+0R=0R, so that d is the real characteristic of R, and d divides t. But we already said that n was the characteristic of R, so this is a contradiction (since d < n and the characteristic is a 'smallest positive integer such that...'). Therefore (in this case) t*a=t*b implies a=b. Note that characteristic infinity presents a minor problem in this case. So suppose that there exist a and b such that t*a=t*b but a != b. Then, as before, we have t*(a-b)=t*c=0R. t is less than infinity, and we again have a contradiction. So such an a and b cannot exist. QED