Let $R$ be an integral domain. If $I \cap J = IJ$ for all ideals $I,J$ of $R,$ then $R$ is a field.

Question

Let $R$ be an integral domain. If $I \cap J = IJ$ for all ideals $I,J$ of $R,$ how do I show $R$ is a field? Hints will suffice. Thank you.

Answer 1

Hint: Let $a\in R\setminus\{0\}$ and consider $I = J = Ra$.

Answer 2

Arithmetically: since ideal intersection $= \rm lcm$ for principal ideals, the hypothesis implies for all $\,a,b\neq 0\,$ we have $\,{\rm lcm}(a,b) = ab,\,$ so $\,\gcd(a,b)=1,\,$ i.e. $\,a,b$ have only unit common factors. Hence for $\,b=a\,$ the common factor $\,a\,$ of $\,a,b$ is a unit, so $\,a\neq 0\,\Rightarrow\, a\,$ is a unit, so $R$ is a field.