Am preparing for exam few days to go. I came across this problem in Anderson - Fuller book about modules.
(1) Give an example of an indecomposable module that has a decomposable submodule.
(2) Give an example of an indecomposable module that has a decomposable factor module.
In part (1), there is a hint : Try a factor module of $R$ (as a left module over itself), where $R = Q[X,Y]$, that is the polynomial ring over rational numbers with two indeterminants. The problem is ... I can't figure out which factor module, and how to describe it since I rarely work with two indeterminants. I'm still working on part (2).
Anyone can give direction? Thanks for the help.
Note : A module $M$ is indecomposable if its direct summands are only $0$ and $M$.