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I am studying for my algebra qualifying exam in January, and this was one of the questions on a recent exam.

Let $R$ be a commutative ring with identity, and let $I$ and $J$ be ideals of $R$. If $I+J=R$, show that $IJ=I \cap J$.

Here is my approach, and I just want to know if it is correct or not.

Suppose $I$ and $J$ are ideals of a commutative ring $R$ such that $I+J=R$. We wants to show that $IJ=I \cap J$. So, we have the following:

$IJ \subset I \cap J$:

Let $x \in IJ$. Then $x=ab$ for some $a \in I$ and $b \in J$. Moreover, $a+b \in I+J$ and so $a+b \in R$. Since $IJ$ can be shown to be an ideal of $R$, then $x(a+b) \in IJ \implies x \in I$, and, since $R$ is commutative, $(a+b)x \in IJ \implies x \in J$. Hence, $x \in I \cap J$, and so $IJ \subset I \cap J$.

$I \cap J \subset IJ$:

Let $x \in I \cap J$. Then, $x \in I$ and $x \in J$. Hence, $xx \in IJ$, and so $x \in IJ$ since $IJ$ is an ideal. Thus, $I \cap J \subset IJ$.

Hence, we have $IJ=I \cap J$.

Bill Dubuque
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Sarah Hadaidi
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    It is incorrect to say that if $x\in IJ$, then $x=ab$ for some $a\in I$, $b\in J$. You should instead say $x=\sum_{k=1}^n a_kb_k$ for some $n\in\mathbb{N}$ and some $a_1,\dots,a_n\in I$, $b_1,\dots,b_n\in J$.

    It's also incorrect to say that $xx\in IJ$ implies $x\in IJ$.

     – Allen Bell Dec 27 '23 at 21:51
  • @AllenBell Since you are answering the question, it is more appropriate to post an answer. Comments should only be used to clarify, not answer the question. See How do comments work for more information. – Martin Brandenburg Dec 27 '23 at 21:59
  • For a solution-verification question to be on topic you must specify precisely which step in the proof you question, and why so. This site is not meant to be used as a proof checking machine. – Bill Dubuque Dec 27 '23 at 23:22
  • @AllenBell, thank you! I think that is what I thought was not quite correct with my proof. – Sarah Hadaidi Dec 28 '23 at 04:34
  • @BillDubuque, I see now why I needed to clarify which step, sorry I am new to the site, so I appreciate your feedback and will keep that in mind. – Sarah Hadaidi Dec 28 '23 at 04:40

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