Equivalent condition of $A/I\cong A/J$.

Question

Let $A$ be a commutative ring, and $I(A)$ be the set of all ideals of $A$.

We define equivalent relation on $I(A)$ by $I\sim J\iff A/I\cong A/J$.

(Remark: We note $A/I\cong A/J$ means ring isomorphic not $A$-algebra isomorphic.)

For example, $A=k[x,y], I=(x), J=(y)$ then $I\neq J$ and $I\sim J. $

Can we state equivalent condition of $I\sim J$ in other words?