Let $R$ be an integral domain and assume that for some non-zero $a \in R$, that $\exists n_a \in \mathbb{N}$ such that $n_a a = 0$. Prove that $R$ has non-zero characteristic.
So here is my thinking so far: I know that every field has characteristic zero or $p$ prime. Since we want to show that $char(R) \neq 0$, can we assume it's prime? If so, I think this is a proof by contradiction. Otherwise, I'm not really sure where to go.
Thanks :)
The repeated sum $n_aa$ is equal to $(n_a1_R)a$. This is a product of two elements of $R$ that is equal to $0$. Hence either $a=0$ or $n_a1_R=0$. Since by hypothesis $a\neq 0$, it follows that $n_a1_R=0$. This means that since there is some $n>0$ for which $n1_R=0$, there in particular exists a smallest one, and this is the characteristic of $R$.