TITLE: Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$
QUESTION [18 upvotes]: In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: 

A lattice is distributive if and only if none of its sublattices is isomorphic to $M_3$ or $N_5$.

where
$\hskip 1.7in$ 
As the answers to the other question demonstrate, it is pretty easy to construct examples of rings with non-distributive ideal lattices; for example, any Noetherian domain either having Krull dimension $>1$, or being non-integrally closed, has a non-distributive ideal lattice. For example, true to the above characterization, the lattice of ideals of $k[x,y]$ has an $M_3$ sublattice:
$$(x,y)$$
$$\text{ / }\qquad |\qquad \text{ \ }$$
$$\quad\quad(x)\quad\quad(y)\,\,\,\,\quad (x+y)$$
$$\text{ \ } \qquad |\qquad \text{ / }$$
$$(0)$$
Of course, it has many other such sublattices, and even in the above example, there are other ideals like $(x+2y)$, $(x^2)$, etc. that live between $(0)$ and $(x,y)$, that we have thrown out for the purposes of getting our $M_3$ sublattice. What I would like to see are examples both of a ring having an ideal lattice isomorphic to $M_3$, and a ring having ideal lattice isomorphic to $N_5$. (I take my rings to be commutative unital rings, and I want to consider the lattice of ideals as including $R=(1)$.)  As I mentioned, there are other ideals between $(0)$ and $(x,y)$, so getting a ring with an $M_3$ ideal lattice is not as easy as localizing $k[x,y]$ at $(x,y)$.
My best shot at producing a ring with an $M_3$ ideal lattice was $R=\mathbb{F}_2[x,y]/(x,y)^2$, which has 
$$R$$
$$|$$
$$(x,y)$$
$$\text{ / }\qquad |\qquad \text{ \ }$$
$$\quad\quad(x)\quad\quad(y)\,\,\,\,\quad (x+y)$$
$$\text{ \ } \qquad |\qquad \text{ / }$$
$$(0)$$
So, what are some good examples of rings having ideal lattice isomorphic to $M_3$? What about $N_5$?

REPLY [7 votes]: I know the original question was about commutative rings, but I just wanted to include in this post that there is a cute example if you allow noncommutative rings. 
In $R=M_2(\mathbb{F}_2)$ (the 2 by 2 matrix ring with entries from the field of two elements) there are exactly five right ideals, and they form the lattice $M_3$. (The same can be said for the left ideals.)
Because $R$ is semisimple each right ideal is generated by an idempotent. One can check there are 6 nontrivial idempotents, but actually it turns out they pair off and they only generate three different ideals. These ideals are:
$$\begin{pmatrix}1&0\\0&0\end{pmatrix}R, \begin{pmatrix}0&0\\0&1\end{pmatrix}R, \begin{pmatrix}0&1\\0&1\end{pmatrix}R 
$$
One should not forget to check that their pairwise intersections are $\{0\}$ and that their pairwise sums are $R$!