## The characteristic of an integral domain is prime or zero

Let $R$ be an integral domain. Prove that the characteristic of $R$ is either prime or zero.

Suppose the characteristic $n$ of $R$ is composite, and that $n = ab$ where $a$ and $b$ are both less than $n$. Letting $\varphi : \mathbb{Z} \rightarrow R$ be the ring homomorphism that takes $k \in \mathbb{Z}$ to the $k$-fold sum of 1 or -1, we have $\varphi(a)$ and $\varphi(b)$ nonzero. However, $\varphi(a)\varphi(b) = \varphi(ab)$ $= \varphi(n) = 0$, so that $\varphi(a)$ and $\varphi(b)$ are zero divisors. Thus we have a contradiction.

Hence, the characteristic of $R$ is not composite, and thus must be a prime or zero.