complete set of primitive orthogonal idempotents
Let be a unital algebra over a field . Recall that is an idempotent iff . If are idempotents, then we will say that they are orthogonal iff . Furthermore an idempotent is called primitive iff cannot be written as a sum where both are nonzero idempotents. An idempotent is called trivial iff it is either or .
Now assume that is an algebra such that
as right modules and for some , . Then , are orthogonal idempotents in and , . Furthermore is indecomposable (as a right module) if and only if is primitive. This can be easily generalized to any number (but finite) of summands.
If is additionally finite-dimensional, then
for some and is a set of pairwise orthogonal, primitive idempotents. This set is called the complete set of primitive orthogonal idempotents of .
|Title||complete set of primitive orthogonal idempotents|
|Date of creation||2013-03-22 19:17:38|
|Last modified on||2013-03-22 19:17:38|
|Last modified by||joking (16130)|