R is a subring of F (field) such that $\forall x \in F$ either $x \in R$ or $x^{-1} \in R$

Question

Suppose $I, J$ are ideals in R, then $I\subseteq J$ or $J \subseteq I$. I tried proving by contradiction but to no avail. Will appreciate any hints.

Answer 1

Suppose that $I$ is not contained in $J$, and pick an $a ∈ I \setminus J$, i.e., with $a\neq 0$. If $b ∈ J$, we must show that $b ∈ I$. If $b = 0$ we are finished, so assume that $b\neq 0$. We have $b/a \in R$, else $a/b ∈ R$, so $a = (a/b)b ∈ J$, a contradiction. Therefore $b = (b/a)a ∈ I$.