Vtyjkyuk

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?

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.

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 $F$, $D_Dsubseteq_e F_D$.

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:

1. In an integral domain, all nonzero ideals are essential.




2. 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.




3. On the other hand: a nontrivial direct summand of $R_R$ is neither essential nor superfluous.




4. 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.