Let $A$ be a unital algebra over a field $k$. Recall that $e\in A$ is an idempotent iff $e^{2}=e$. If $e_{1},e_{2}\in A$ are idempotents, then we will say that they are orthogonal iff $e_{1}e_{2}=e_{2}e_{1}=0$. Furthermore an idempotent $e\in A$ is called primitive iff $e$ cannot be written as a sum $e=e_{1}+e_{2}$ where both $e_{1},e_{2}\in A$ are nonzero idempotents. An idempotent is called trivial iff it is either $0$ or $1$.

Now assume that $A$ is an algebra such that

as right modules and $1=m_{1}+m_{2}$ for some $m_{1}\in M_{1}$, $m_{2}\in M_{2}$. Then $m_{1}$, $m_{2}$ are orthogonal idempotents in $A$ and $M_{1}=m_{1}A$, $M_{2}=m_{2}A$. Furthermore $M_{i}$ is indecomposable (as a right module) if and only if $m_{i}$ is primitive. This can be easily generalized to any number (but finite) of summands.

If $A$ is additionally finite-dimensional, then

for some (unique up to isomorphism) right (ideals) indecomposable modules $P_{i}$. It follows from the preceding that

for some $e_{i}\in A$ and $\{e_{1},\ldots,e_{n}\}$ is a set of pairwise orthogonal, primitive idempotents. This set is called the complete set of primitive orthogonal idempotents of $A$.