For two right modules is an essential submodule of (that is ) iff for any submodule Equivalently, it's an essential submodule iff for any there is such that
The singular submodule of an module is the set of all such that A module is singular iff
I have to prove that if an module is singular, then there are two modules such that
I have no idea how to go about it. I'm completely new to module theory and I don't have much intuition regarding it. How does one find two modules whose quotient is given but which have to satisfy a certain condition?
I was thinking this:
Let's take any two modules ( ) such that Let and let be the equivalence class of Then there is such that That means that which in turn means that But this proves nothing, because the important part is that I do know that but this is not enough.