Show that the characteristic of an integral domain must be either zero or a prime.
Solution
Let \(R\) be an integral domain, so \(1_R\ne0_R\) and \(R\) has no zero divisors, and let \(n=\operatorname{char}(R)\): either \(n=0\), or \(n\) is the least positive integer with \(n\cdot1_R=0_R\).
1. If \(n>0\), then \(n\ge2\).
\(1\cdot1_R=1_R\ne0_R\).
2. If \(n>0\), then \(n\) is prime.
Suppose \(n=ab\) with integers \(1<a,b<n\). Then \((a\cdot1_R)(b\cdot1_R)=(ab)\cdot1_R=n\cdot1_R=0_R\), so \(a\cdot1_R=0_R\) or \(b\cdot1_R=0_R\) because \(R\) has no zero divisors. Either equality contradicts the minimality of \(n\). Hence \(n\), which is at least \(2\) by step 1, has no factorization into smaller integers greater than \(1\), so \(n\) is prime.
3. Q.E.D.
Either \(n=0\), or \(n>0\) and \(n\) is prime by step 2.