2 changed to eplicit (ugly but working...) links

In Hungerford everything is defined without assuming that rings have identity (or that they are commutative). At least according to Definition IV.1.3 on p. 171 of Hungerford, the answer to your first question is affirmative. Quoting:

Definition 1.3. Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$. $B$ is a submodule of $A$ provided that $B$ is an additive subgroup of $A$ and $rb\in B$ for all $r\in R$, $b\in B$.

[I hope that my interpretation of the question was correct. I understood 'characterization of submodule' as 'definition of submodule.']

EDIT: Google books links to the definition of a module, a submodule, and a ring in Hungerford:

Definition 1.3. Let $R$ be a ring, $A$ an $R$-module and $B$ a nonempty subset of $A$. $B$ is a submodule of $A$ provided that $B$ is an additive subgroup of $A$ and $rb\in B$ for all $r\in R$, $b\in B$.