Question

3.7 If D is an integral domain and if na = 0 for some a #= 0 in

D and some integer n #= 0, prove that D is of finite characteristic.

Answer 1

Let $b\in D$. Consider $nab$. Since $(na)=0$ we have $nab=0$. On the other hand, $nab=nba$ as the integral domain is a commutative ring. So $0=nab=nba$. But $a\ne 0$ implies $nb=0$ for all $b\in D$.