Let $R$ be a ring. The definition of essential/superfluous submodules of some $R$-module is pretty simple to understand, but I'm wondering about essential (no pun intended) examples for both notions.
What are the examples I should keep in my head when I hear about the essential/superfluous modules?
On
These two things are pretty practical illustrations:
In any ring (with identity) the Jacobson radical is a superfluous submodule (on either side.)
For any commutative integral domain $D$ with field of fractions $F$, $D_D\subseteq_e F_D$.
Doh, I was a little too slow with the above two, and Christopher has already covered them. I'll try to include some other things.
Some other observations:
In an integral domain, all nonzero ideals are essential.
Consider any ring in which the right ideals are linearly ordered (such as a valuation domain.) . Then all nontrivial ideals are both essential and superfluous.
On the other hand: a nontrivial direct summand of $R_R$ is neither essential nor superfluous.
A ring has no proper essential right ideal iff it is a semisimple (Artinian) ring. A ring has no proper superfluous right ideal iff it has trivial Jacobson radical. Notice that the first observation implies the second.
The classic example of an essential submodule is $\Bbb Z \subset \Bbb Q$ (considered as $\Bbb Z$-modules), or more generally a domain $R$ considered as a submodule of its field of fractions (again as $R$-modules).
The classic example of a superfluous submodule is the Jacobson radical $J(R)$ of a ring $R$ (considered as a submodule of $R$), or more generally $J(R)M \subset M$ for any finitely generated $R$-module $M$. See Rotman for more details.