# Proof verification: the characteristic of an integral domain $D$ must be either 0 or prime.

Claim: the characteristic of an integral domain $$D$$ must be either 0 or prime.

Here is my attempt: Assume $$D$$ is an integral domain. Assume $$k$$ is the characteristic of $$D$$. Let $$a \in D\setminus \{0\}$$. Aiming for a contradiction, assume $$k$$ is neither prime nor $$0$$. Since $$k$$ is the smallest positive integer satisfying $$k \cdotp a = 0$$, $$\exists m, n \in \mathbb{Z}^+$$ s.t. $$\begin{equation} k = m \cdotp n \end{equation}$$ Without loss of generality, assume that $$m, n$$ are the smallest positive integers satisfying $$k = m \cdotp n$$. Since $$D$$ is a ring with unity $$1 \neq 0$$, we have $$k = (m \cdotp 1) \cdotp (n \cdotp 1)$$. That is, $$(m \cdotp 1) \cdotp (n \cdotp 1) \cdotp a= 0$$. Since $$D$$ contains no divisors of $$0$$, either $$(m \cdotp 1) = 0$$ or $$(n \cdotp 1) = 0$$. If $$(m \cdotp 1) = 0$$, then by Theorem 19.15, $$n$$ is the characteristic of $$D$$ is $$n$$, which is a contradiction. If $$(n \cdotp 1) = 0$$, then by Theorem 19.15 again, the characteristic of $$D$$ is $$m$$, which is also a contradiction. $$\square$$

Theorem 19.15: Let $$R$$ be a ring with unity. If $$n \cdotp 1 = 0$$ for some $$n \in \mathbb{Z}^+$$, then the smallest such integer $$n$$ is the characteristic of $$R$$.

My question: I am not sure if my use of Theorem 19.15 is correct/ justified in my proof. I know that I have "Without loss of generality, assume that $$m, n$$ are the smallest positive integers satisfying $$k = m \cdotp n$$" in my proof but I am not sure if this is sufficient to use Theorem 19.15 the way I have in the last couple lines of my proof.

Can someone please verify if this proof is correct or if it needs any adjustments? Thanks!

## 1 Answer

Yes, it's all correct, though you don't need $$n$$ to be the smallest nontrivial divisor. (Note that saying the pair $$n,m$$ is 'smallest' makes no sense.)

Theorem 19.15 can be easily seen, as if $$n\cdot 1=0$$, then $$n\cdot a=(n\cdot 1)\cdot a=0$$ for every element $$a$$.

• Thanks! Can you please explain how Theorem 19.15 is "easily seen" using the equation you have in the answer? What is $a$ in your equation? How does it show that $n$ is the smallest integer satisfying $n \cdotp 1 = 0$. – Ricky_Nelson May 24 '20 at 0:29
• @Ricky_Nelson We have $(\forall a:na=0)\iff (n1=0), and the definition of characteristic is the smallest positive$n\$ satisfying the left side. – Berci May 24 '20 at 9:49