What is the simplest non-principal ideal? Let's restrict ourselves to commutative rings (not necessarily with unity).
Is there a simpler example of a non-principal ideal than $\langle a,x\rangle$ in $R[x]$, where $a\in R$ is not a unit (and therefore $R$ is not a field)? All other examples that come to mind involve more complicated polynomial rings and seem to be particular cases of the previous example.
 A: The classic example for a ring that is not a polynomial ring  is the ideal $(2,1+\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$. I don't think there's going to be anything simpler if you rule out examples from polynomial rings.
But personally I find the conceptually simplest example is the ideal $(x,y)$ in $R[x,y]$, where $R$ can be any non-zero ring at all. It requires the least amount of thought to see that it is non-principal. However it fits into the pattern you've ruled out, since $R[x,y]=R[x][y]$.
A: This is an answer to the comment of dafnahaktana above to the answer

How do prove that $(2,1+\sqrt{-5})$ is non principal in $\mathbb{Z}[\sqrt{-5}]$ ? 

Please inform him, I have not enough reputation to add a comment.
The ideal $(2,1+\sqrt{-5})$ in the commutative ring $\mathbb{Z}[\sqrt{-5}]$ is
\begin{align*}
&(2,1+\sqrt{-5})\\
=&\{2r_1+(1+\sqrt{-5})r_2:r_1,r_2\in \mathbb{Z}[\sqrt{-5}] \}\\
=&\{2(r_{11}+r_{12}\sqrt{-5})+(1+\sqrt{-5})(r_{21}+r_{22}\sqrt{-5}):r_{11},r_{12},r_{21},r_{22}\in \mathbb{Z} \}\\
=&\left\{ \left(2r_{11}+r_{21}-5r_{22}\right) +\left(2r_{12}+r_{21}+r_{22}\right)\sqrt{-5} :r_{11},r_{12},r_{21},r_{22}\in \mathbb{Z} \right\}
\end{align*}
Which elements of $\mathbb{Z}[\sqrt{-5}]$ are in $(2,1+\sqrt{-5})$? Let us solve the system
\begin{gather*}
\begin{cases}2r_{11}+r_{21}-5r_{22}=n_1\\
2r_{12}+r_{21}+r_{22}=n_2\end{cases} 
\end{gather*}
If $n_1$ and $n_2$ have the same oddity, we can choose any $r_{22}$ an choose $r_{21}$ in such a way that $r_{21}-5r_{22}$ and $r_{21}+r_{22}$ have also the same oddity, then the values of $r_{11}$ and $r_{12}$ follow; if $n_1$ and $n_2$ have opposite oddity, then there is no solution, thus
\begin{gather*}
(2,1+\sqrt{-5})=\{n_1+n_2\sqrt{-5}:n_1,n_2\in\mathbb{Z}\text{ and $n_1,n_2$ with equal oddity}\}.
\end{gather*}
Thus the lattice/grid of $(n_1,n_2)$ in $\mathbb{Z}^2$ cannot be generated by a single "vector".
