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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.

drhab
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  • if $R$ is not commutative, then require $I$ and $J$ to be two-sided ideals. – lhf Jun 04 '14 at 19:19
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    @lhf $I$ and $J$ are indeed two-sided ideals. But how does that help? – drhab Jun 04 '14 at 21:06
  • Related: https://math.stackexchange.com/questions/1222474/assume-r-is-commutative-if-ij-r-prove-that-ij-i-cap-j-provide-a-count?noredirect=1&lq=1 – Watson Dec 22 '16 at 17:45

1 Answers1

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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.

drhab
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