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.
Best Answer
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 $R[x,y]=R[x][y]$.