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.

  • $\begingroup$ if $R$ is not commutative, then require $I$ and $J$ to be two-sided ideals. $\endgroup$
    – lhf
    Jun 4, 2014 at 19:19
  • 2
    $\begingroup$ @lhf $I$ and $J$ are indeed two-sided ideals. But how does that help? $\endgroup$
    – drhab
    Jun 4, 2014 at 21:06
  • $\begingroup$ Related: math.stackexchange.com/questions/1222474/… $\endgroup$
    – Watson
    Dec 22, 2016 at 17:45

1 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.


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