TITLE: Non-unital module over a ring with identity?
QUESTION [9 upvotes]: Is there a nontrivial example of a non-unital module over a ring with identity?  By trivial, I mean modules  with $rm = 0$ for all $r$ and $m$.
Just an idle question.  (Why is there no tag for that?)

REPLY [13 votes]: In essence there is no counterexample because of the following lemma:

Lemma: Every (not necessarily unital) module is a direct sum of a unital module and a trivial module.

Proof: Let $V$ be the module. Then we claim that $V=1_AV\oplus \operatorname{ker}\lambda_{1_A}$, where  $\lambda_{1_A}$ is left multiplication with $1_A$. Indeed let $v$ be in their intersection, then $v=1_Aw$ and $0=1_Av=1_A^2w=1_Aw=v$, hence the intersection is zero and furthermore $v=1_Av+(v-1_Av)$ with the former summand belonging to $1_AV$, the latter to $\operatorname{ker}\lambda_{1_A}$.