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 $A$ .

Title	complete set of primitive orthogonal idempotents
Canonical name	CompleteSetOfPrimitiveOrthogonalIdempotents
Date of creation	2013-03-22 19:17:38
Last modified on	2013-03-22 19:17:38
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Definition
Classification	msc 16S99
Classification	msc 20C99
Classification	msc 13B99