Operations with ideals in a commutative ring

Question

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?

Answer 1

We have $(A\cap B)(A+B)\subseteq AB$, but the other containment isn't always true.

In the ring $K[X,Y]$, where $K$ is a field and $X$ and $Y$ are indeterminates, we have

$$\Big((X)\cap(Y)\Big)\Big((X)+(Y)\Big)\subsetneq(X)(Y).$$ (Note that $(X)\cap(Y)=(XY)$, and therefore $(XY)(X,Y)\subsetneq(XY)$; otherwise $1\in(X,Y)$, a contradiction.)