An a unital ring , an idempotent is called a division idempotent if , with the product of , forms a division ring. If instead is a local ring – here this means a ring with a unique maximal ideal where a division ring – then is called a local idempotent.
Suppose with and . Then by cancellation . ∎
The integers are an integral domain which is not a division ring and they serve as a counter-example to many conjectures about idempotents of general rings as we will explore below. However, the first important result is to show the hierarchy of idempotents.
Every local ring has only trivial idempotents and .
Let be the unique maximal ideal of . Then is the Jacobson radical of . Now suppose is an idempotent. Then must be left invertible (following the element characterization of Jacobson radicals (http://planetmath.org/JacobsonRadical)). So there exists some such that . However, this produces
Thus every non-trivial idempotent lies outside . As is a division ring, the only idempotents are and . Thus if , is an idempotent then it projects to an idempotent of and as it follows projects onto so that for some . As we find (often called an anti-idempotent). Once again as we know there exists a such that and so indeed . ∎
Every division idempotent is a local idempotent, and every local idempotent is a primitive idempotent.
Let be a unital ring. Then in the standard idempotents are the matrices
If has only trivial idempotents (i.e.: and ) then each is a primitive idempotent of .
If is a local ring then each is a local idempotent.
If is a division ring then each is a division idempotent.
When then (i) is not satisfied and consequently neither are (ii) and (iii). When then (i) is satisfied but not (ii) nor (iii). When – the formal power series ring over – then (i) and (ii) are satisfied but not (iii). Finally when then all three are satisfied.
A consequence of the Wedderburn-Artin theorems classifies all Artinian simple rings as matrix rings over a division ring. Thus the primitive idempotents of an Artinian ring are all local idempotents. Without the Artinian assumption this may fail as we have already seen with .
|Date of creation||2013-03-22 16:48:43|
|Last modified on||2013-03-22 16:48:43|
|Last modified by||Algeboy (12884)|