TITLE: What is the example of a module that is local but not endolocal?
QUESTION [5 upvotes]: A module $M$ over a ring $R$ is called local if $M$ has a largest proper submodule. 
A module $M$ over a ring $R$ is called endolocal if $End_R(M)$ is local.
I am trying to find an example for a local module but not endolocal.

REPLY [2 votes]: If $M$ is a uniserial right $R$-module that is neither hopfian nor cohopfian, then $End(M_R)$ has exactly two maximal left ideals. $M$ is of course local while this endomorphism ring is not.
You can find more details in Lam's First course in noncommutative rings in the appendix on such endomorphism rings.