Does the definition of "local module" need to say all proper submodules are contained in the maximal submodule? I would like to know: Is there any definition for local modules  like definition  for local rings?. In other words: what is the name of a module with only one nontrivial maximal submodule?
 A: I think this is a natural question, since the definition for rings does not encounter this subtlety.  The question is which definition to use:


*

*A module with exactly one maximal submodule


*A module with a proper submodule containing all other proper submodules (a.k.a. a 'greatest' proper submodule)

The second one implies the first, of course, but the converse is not obvious. You can make a module with maximal submodules and submodules not contained in any maximal submodule: just take a module $M$ with no maximal submodules, and then form the direct product with a simple module over the same ring. This leaves room for some doubt that the two definitions of the same.
On page 130 of Ring and Module Theory by Albu, Birkenmeier, Erdogan & Tercan, they give it more explicitly (paraphrased):

A local module is a hollow module such that $Rad(M)\neq M$.

Hollow means that every proper submodule is small, and of course a small submodule $S$ of $M$ is one for which $S+N=M$ implies $N=M$. The last half of the definition says that a maximal submodule (call it $B$) exists. Then if $S$ is any other proper submodule, $S+B$ is either $B$ or $M$. If it is $M$, smallness of $S$ implies that $B=M$, a contradiction. Therefore $S+B=B$ and so $S\subseteq B$.
Conversely, it's easy to see that a module with a proper submodule containing all other proper submodules is necessarily hollow and $Rad(M)\neq M$.  If $N$ and $N'$ are both proper submodules, then there is no way $N+N'=M$ since they are both contained in the maximal submodule.
Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules p 5 by Facchini says:

Has a greatest proper submodule. Equivalently, it is cyclic, nonzero and has a unique maximal submodule.

The last half solves the problem this way: if the module is cyclic, it's just a quotient of $R$, and then correspondence gives us that submodules are contained in maximal submodules. The first half, by saying "greatest" seems to include the clause that all proper submodules are contained in the maximal submodule also.
Injective Modules and Injective Quotient Rings by Faith, p 26

Has a unique maximal submodule. Equivalently $M/rad(M)$ is simple.

Faith does not appear to rule out submodules not contained in maximal submodules.
Serial rings by Puninski, p 6 is apparently a little garbled (this is verbatim:)

A module $M$ is called local if $M$ contains the largest proper submodule.

Again, this could be interpreted as "contains the" meaning "has a" and that "largest" entails containment of all proper submodules.

In summary, I'm not completely sure there exists an example of a module with one maximal submodule and submodules that aren't contained in maximal submodules. It seems like there is a small gap in between the two definitions at the top of this solution, but I have no example.
I think the clearest definition is to say something explicitly that requires the unique maximal submodule contain all other proper submodules. It feels like there could be pathological behavior otherwise.
