complete set of primitive orthogonal idempotents

Let A be a unital algebra over a field k. Recall that eA is an idempotentPlanetmathPlanetmath iff e2=e. If e1,e2A are idempotents, then we will say that they are orthogonalMathworldPlanetmathPlanetmathPlanetmath iff e1e2=e2e1=0. Furthermore an idempotent eA is called primitivePlanetmathPlanetmath iff e cannot be written as a sum e=e1+e2 where both e1,e2A are nonzero idempotents. An idempotent is called trivial iff it is either 0 or 1.

Now assume that A is an algebraPlanetmathPlanetmath such that


as right modules and 1=m1+m2 for some m1M1, m2M2. Then m1, m2 are orthogonal idempotents in A and M1=m1A, M2=m2A. Furthermore Mi is indecomposableMathworldPlanetmath (as a right module) if and only if mi 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 isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) right (ideals) indecomposable modules Pi. It follows from the preceding that


for some eiA and {e1,,en} 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