I want an example of a non noetherian commutative ring with unity which has a nonzero ideal which is contained in every other ideal.
The following are commutative non-Noetherian rings with simple, essential socles. This amounts to the intersection of nonzero ideals being nonzero.
$ \mathbb F_2[x_1, x_2, x_3,\ldots ]/(\{x_i^3\mid i\in \mathbb N\}\cup\{x_ix_j\mid i\neq j\}\cup\{x_i^2-x_j^2\mid i,j\in \mathbb N\}),$ where $\mathbb F_2$ is the field of two elements.
The trivial extension $D(+)V$ where $D=F[[x]]$ for a field $F$, and $V=F((x))/D$
The trivial extension $\mathbb Z (+) \mathbb Z_{p^\infty}$ of a Prüfer $p$-group by the integers.
The monoid ring $\mathbb F_2[M]$ where $\mathbb F_2$ is the field of two elements and $M$ is the monoid structure on the interval $[0,1]$ where
$$ab:=
\begin{cases}
a+b & \text{ if } a+b \leq 1 \\
0 & \text{ otherwise }
\end{cases}.
$$
The ideals of the second and fourth one are actually linearly ordered. The first and fourth one are not useful to you because they are local with nil radical, so $\{0\}$ is primary.
To see $\{0\}$ is not primary in the third one, take an element of the Prüfer group of order $p$, say $x$, then $(0,x)(p,0)=0$, and yet $(0,x)$ is not zero, and $(p,0)$ is not nilpotent.
The second one will also work for you, for similar reasons. You can use $a=(0, x^{-1}+k[[x]])$ as a nonzero element, and $b=(x,0)$ as a non-nilpotent element, and $ab=0$.
Clearly you can see the trivial extension construction is useful for creating non-primary zero ideals!
I'm using $(+)$ to denote the trivial extension. In detail, if we have a ring $R$ and a module $M$, the underlying set of $R(+)M$ is $R\times M$ and the operations are coordinatewise addition, and multiplication that is $(a,m)(b,n):=(ab,an+mb)$. Writing it as $R(+)M$ is just a way to make sure it's not confused for the product ring.
