Hw12 Problem 14

Problem: (*) Show that the characteristic of an integral domain must be either $0$ or prime.

Solution: Using Theorem 19.15 we can look solely at $mn * 1 = 0$ for $m > 1$ and $n > 1$, where $mn$ is the characteristic of the integral domain. Now,

(1)
$$mn * 1 = (m * 1)(n * 1)$$
(2)
$$= 0$$

Since we are operating within an integral domain, either $m * 1 = 0$ or $n * 1 = 0$. So then the characteristic is both at most $m$ and at most $n$. Then the characteristic cannot be a composite integer. Therefore the characteristic of an integral domain is either $0$ or prime.