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.

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