Do $IJ$ and $I\cap J$ coincide if $I$ and $J$ are coprime? Also if ring $R$ has a $1$ and is not commutative?

Question

Let $R$ be a ring (with identity) and let $I,J$ be two coprime (two-sided) ideals in it.

In Algebra: Chapter $0$, Aluffi, III. exercise 4.5.

the reader is asked to prove that:

$$IJ=I\cap J$$

I have the following proof for $IJ+JI=I\cap J$:

It is evident that $IJ+JI\subset I\cap J$, and if $i+j=1$ with $i\in I$ and $j\in J$ then for $a\in I\cap J$ we have: $a=ia+ja\in IJ+JI$.

So I would be ready if $R$ is commutative, but that is not one of the data.

Can you help me with a proof or counterexample?

Thanks in advance and sorry if this is a duplicate.

Answer 1

I have had a look on this where errors in the book mentioned in the question are exposed. The ring should be a commutative one after all. In that case $$IJ+JI=IJ$$ so my proof is complete.

I am not really interested in a counterexample when it comes to rings that are not commutative.