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单词 CompleteSetOfPrimitiveOrthogonalIdempotents
释义

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

A=M1M2

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

A=P1Pn

for some (unique up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) right (ideals) indecomposable modules Pi. It follows from the preceding that

Pi=eiA

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.

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更新时间:2025/5/28 11:48:47