Examples for essential/superfluous submodules 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.