A ring $R$ is said to have "enough idempotents" if there exists some set of idempotents $E = \{e_i ~|~ i \in I\}$ such that $_RR \simeq \bigoplus_{i \in I} Re_i$ as left $R$-modules.
This is related to the property that an algebra has a set of local units. A set of idempotents $U$ is a set of local units if for every finite subset $\{a_1, a_2, \ldots, a_n\} \subseteq R$, there exists $e \in U$ such that $a_i e = e a_i = a_i$.
It is straightforward to show that $R$ having enough idempotents implies that $R$ has a set of local units. Under what conditions does the converse hold?