complete set of primitive orthogonal idempotents

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

A=M_{1}\\oplus M_{2}

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

A=P_{1}\\oplus\\cdots\\oplus P_{n}

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

P_{i}=e_{i}A

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 $A$ .