What can be said of the lattices of submodules of a noetherian/finitely generated module?

Question

I've asked several times about properties of the lattice of submodules/ideals of modules/rings with specific properties.

This time I wonder about what interesting properties can one see in the lattice of submodules of a noetherian module and a finitely generated module.

For instance I read Lasker-Noether theorem which states (in the module formulation):

Every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules.

In general, I don't have the intuition to foresee what will be the properties of these lattices. Let me give a preliminary list in any case to try to be more concrete.

Properties of lattices (wikipedia's classification of lattices)

ok: Complete, Modular,Algebraic lattice,Arguesian lattice

no: Distributive, Complemented

Metric lattice ??

Projective lattice ??

...

So do the lattices of submodules of noetherian/finitely generated modules verify any of the above properties? Is there something else interesting that can be said about then?

Any references are welcome.

Another way of looking at the problem:

Given the lattice of submodules $L(M)$ of a module $M$, how to know if it is noetherian/finitely generated?

M is Noetherian $\iff$ every ascending chain of submodules stabilizes.

$M$ finitely generated $\iff$ every ascending chain of submodules with union M stabilizes.

Answer 1

I can strengthen both of rschwieb's properties:

The lattice of submodules is always Arguesian and algebraic.

This is a strengthening since Arguesian lattices are modular and algebraic lattices are complete.

An Arguesian lattice is a lattice satisfying a complicated identity that comes from projective geometry. The lattice of subspaces of a projective space is Arguesian iff the space satisfies Desargue's law. Since the lattice of subspaces is always modular, there exist modular non-Arguesian lattices.

An algebraic lattice is a complete lattice in which every element is the join of compact elements. Here is an example of a complete lattice that is not algebraic.

Answer 2

The lattice of submodules of any module over any ring is modular and complete.

The lattice of a finitely generated/Noetherian module needs not be distributive. Example: Let $F$ be a field (finite, if you like) and $M=F\times F$ be a vector space of dimension $2$. The lattice of submodules has the diamond lattice, which is not distributive.

The lattice need not be complemented. Example: let $F$ be a field (finite, if you like), and $R=F[x]/(x^2)=M$. There is precisely one nontrivial submodule and it is not complemented.