Theorem 19.15

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Statement:

Let R be a ring with unity. If $n \cdot 1\not= 0$ for all $n\in \mathbb{Z^+}$, then R has characteristic 0. If $n \cdot 1\ = 0$ for some $n\in \mathbb{Z^+}$, then the smallest such integer n is the characteristic of R.

Proof:
If $n \cdot 1\not= 0$ for all $n\in \mathbb{Z^+}$, then surely we cannot have $n \cdot a= 0$ for all $a\in \mathbb{R}$ for some positive integer n, so by Definition 19.13, R has characteristic 0.
Suppose that n is a positive integer such that $n \cdot 1= 0$. Then for any $a\in \mathbb{R}$, we have $n \cdot a= a+a+a+\dots +a=a(1+1+\dots +1)=a(n\cdot 1)=a0=0$.