 Orthogonal idempotent

A finite family of idempotents $e_1,\dots ,e_r$ in a unital ring $A$ is orthogonal whenSuppose also that $e_1+\cdots +e_r=1$. For a unital left module $M$, these elements give a decomposition of its underlying abelian group:The decomposition exists because $m={\sum }_i{e}_{i}m$. It is unique because applying $e_j$ to a relation ${\sum }_i{m}_i=0$, with $m_i\in e_{i}M$, isolates $m_j=0$. This argument works for modules over general rings, without requiring an underlying vector space.

If every $e_i$ is central, each $e_{i}M$ is also an $A$-submodule, since $a(e_{i}m)=e_i(am)$. For a noncentral example, take $A=M_2(k)$ acting on $M=k^2$ and the matrix units $E_{11},E_{22}$. Their images are the two coordinate axes. The matrix unit $E_{12}$ sends $(0,1)$ to $(1,0)$, so the second axis is not an $A$-submodule. Over a field the decomposition is still a vector space decomposition. The vertex idempotents of a path algebra similarly supply the separate spaces in a quiver representation.