# 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 iff $e_{1}e_{2}=e_{2}e_{1}=0$. Furthermore an idempotent $e\in A$ is called 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 $A$.

Title complete set of primitive orthogonal idempotents CompleteSetOfPrimitiveOrthogonalIdempotents 2013-03-22 19:17:38 2013-03-22 19:17:38 joking (16130) joking (16130) 4 joking (16130) Definition msc 16S99 msc 20C99 msc 13B99