Let $I$ and $J$ be two ideals. Then is $IJ=JI$?

Question

In a commutative ring $A$, $IJ=JI$ where $I$ and $J$ are two ideals of $A$.

But if $A$ is not commutative is it true?? $IJ$ is the product of ideals $I$ and $J$.

Answer 1

Hint:

One might try something obvious like $R=\mathbb Z\langle x,y\rangle/(x^2, xy, y^2)$.

Let $I=(x)$ and $J=(y)$ and compute what these ideals look like and what their products look like, in both orders.