Why “characteristic zero” and not “infinite characteristic”? 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. 

Given a ring $R$ there is a unique ring homomorphism $\varphi:\mathbb Z\to R$. The characteristic of $R$ is the (canonical, non-negative) generator of $\ker \varphi$.
