TITLE: Why “characteristic zero” and not “infinite characteristic”?
QUESTION [123 upvotes]: The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we define it to be $0$. 
But why characteristic zero? Why do we not define it to be $\infty$ instead? Under this alternative definition, the characteristic of a ring is simply the “order” of the additive cyclic group generated by the unit element $1$. 
My feeling is that there is a precise and convincing explanation for the common convention, but none comes to mind.  I couldn't find the answer in the Wikipedia article either.

REPLY [2 votes]: This is all just a convention, and I'm adding an answer $8$ years late because nobody else has pointed out yet that many years ago some people did use the term "characteristic $\infty$". Schilling does this in his book "Theory of Valuations" (AMS, 1950). Either way, you get used to it.