When $I+J$ is a special indecomposable ideal of $R$

Question

$\newcommand{\Ann}{\operatorname{Ann}}\newcommand{\Max}{\operatorname{Max}}$I am looking for an example of a commutatvive ring $R$ with $1$ having two ideals $I$ and $J$ such that $I\cap J\not=0$, $\sqrt{\Ann(I)}, \sqrt{\Ann(J)}\in \Max(R)$, and $I+J$ is indecomposable ideal of $R$.

Where, $I+J$ is indecomposable if it is not a direct sum of two nonzero ideals, $\Ann(T):=\{r\in R\mid rT=0\}$ for $T\subseteq R$, and $\Max(R)$ is the set of all maximal ideals of $R$.

Answer 1

$\newcommand{\Ann}{\operatorname{Ann}}$Let $A=K[x]/(x^n)$ and $I=\langle x^a\rangle$ and $J=\langle x^b\rangle$ for $1 \leq a \leq b \leq n-1$. Their intersection is nonzero. $\Ann(I)=\langle x^{n-a}\rangle, \Ann(J)=\langle x^{n-b}\rangle$ and their radicals are $\langle x^1\rangle$, which is maximal. $I+J=\langle x^{\min(a,b)}\rangle$ is indecomposable.