I want to know if a simple ( minimal) ideal of a commutative ring with ideantity is contained in the sum of some ideal we can deduced it is contained in one of them.
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$\begingroup$Nope.
Take $R=\left\{\begin{bmatrix}a&b\\0&a\end{bmatrix}\middle|a\in\mathbb R, b\in\mathbb R^2 \right\}$
The ideals generated by $\begin{bmatrix}0&(1,1)\\0&0\end{bmatrix}$, $\begin{bmatrix}0&(0,1)\\0&0\end{bmatrix}$ and $\begin{bmatrix}0&(1,0)\\0&0\end{bmatrix}$ are all minimal, and the sum of each pair contains the third, but none of them mutually contain each other.
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