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