In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar result is true. Is it true in modules over a nonunital ring?


I should mention that the analogue of the following is not true. 


In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if for all scalars $\alpha\_1, \alpha\_2$ and for all $w\_1, w\_2 \in W$, $\alpha\_1 w\_1 + \alpha\_2 w\_2 \in W$. 


This can be seen by the module $\mathbb{Z}$ over $2\mathbb{Z}$ and the subset $2\mathbb{Z} \cup \{-3,3\}$.



characterization of a submodule
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: 


<http://books.google.com/books?id=t6N_tOQhafoC&lpg=PP1&dq=hungerford&hl=iw&pg=PA169#v=onepage&q=&f=false>


<http://books.google.com/books?id=t6N_tOQhafoC&lpg=PP1&dq=hungerford&hl=iw&pg=PA171#v=onepage&q=&f=false>


<http://books.google.com/books?id=t6N_tOQhafoC&lpg=PP1&dq=hungerford&hl=iw&pg=PA115#v=onepage&q=&f=false>


