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As the title goes, I am wondering if we have the property that $AB$ equals $(A+B)(A\cap B)$ for ideals $A,B$ of a commutative ring $R$.

I came up with this question because I am recently reading about Dedekind domains in algebraic number theory and one can view the ideals $A+B$ and $A\cap B$ as $\text{lcm}(A,B)$ and $\gcd(A,B)$ respectively. In $\mathbb{Z}$, we have $mn = \text{lcm}(m,n)\gcd(m,n)$, and that's why I wonder if we have this analogous result in the language of ideals.

The following is what I've got so far: For $\sum(a_i+b_i)c_i \in (A+B)(A\cap B)$ with $a_i \in A$, $b_i\in B$, and $c_i\in A\cap B$, each $a_ic_i$ lies in $AB$, and so does $b_ic_i$ ($\because R$ is comm.), so $\sum(a_i+b_i)c_i \in AB$.

Then I get stuck with the reverse. Does the reverse hold for any commutative $R$? Dedekind $R$? Or?

Thanks in advance for giving any clue~

DML
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    I was going through the "Related" section and found out that there is a nice and complete answer by Matt E here: https://math.stackexchange.com/a/263034/588038. – cqfd Aug 05 '21 at 16:41

1 Answers1

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Let $R=k[x,y]$ and $\mathfrak a=(x),\mathfrak b=(y)$ be ideals of $R$. Then $\mathfrak{a}\mathfrak{b}=(xy)$, $\mathfrak a+\mathfrak b=(x,y)$ and $\mathfrak a\cap\mathfrak b=(xy)$. Clearly, $xy\notin (\mathfrak a+\mathfrak b)(\mathfrak{a}\cap\mathfrak{b})=(x^2y,xy^2)$.

The converse holds for Dedekind domains. See corollary 4.5 here: https://kconrad.math.uconn.edu/blurbs/gradnumthy/idealfactor.pdf.

cqfd
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