PlanetMath: idempotent classifications

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idempotent classifications

(Definition)

An a unital ring , an idempotent $e\in R$ is called a division idempotent if $eRe=\{ere:r\in R\}$ , with the product of , forms a division ring. If instead is a local ring - here this means a ring with a unique maximal ideal $\mathfrak{m}$ where $eRe/\mathfrak{m}$ a division ring - then is called a local idempotent.

Lemma 1 Any integral domain has only the trivial idempotents 0 and . In particular, every division ring has only trivial idempotents.

Proof. Suppose $e\in R$ with $e\neq 0$ and

. Then by cancellation

. $\qedsymbol$

The integers are an integral domain which is not a division ring and they serve as a counter-exmple 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.

Theorem 2 Every local ring has only trivial idempotents 0 and .

Proof. Let $\mathfrak{m}$ be the unique maximal ideal of

. Then $\mathfrak{m}$ is the Jacobson radical of

. Now suppose $e\in \mathfrak{m}$ is an idempotent. Then

must be left invertible (following the element characterization of Jacobson radicals). So there exists some $u\in R$ such that

. However, this produces

$\displaystyle e=u(1-e)e=u(e-e^2)=u(e-e)=0.$

Thus every non-trivial idempotent $e\in R$ lies outside $\mathfrak{m}$ . As $R/\mathfrak{m}$ is a division ring, the only idempotents are 0 and

. Thus if $e\in R$ , $e\neq 0$ is an idempotent then it projects to an idempotent of $R/\mathfrak{m}$ and as $e\notin \mathfrak{m}$ it follows

projects onto

so that

for some $z\in\mathfrak{m}$ . As

we find

(often called an anti-idempotent). Once again as $z\in\mathfrak{m}$ we know there exists a $u\in R$ such that

and

so indeed

. $\qedsymbol$

Corollary 3 Every division idempotent is a local idempotent, and every local idempotent is a primitive idempotent.

Example 4 Let be a unital ring. Then in the standard idempotents are the matrices

$\displaystyle E_{ii}=\begin{bmatrix} 0 & \ & \ddots & \ & & 1 \ & & & \ddots\ & & & & 0 \end{bmatrix},\qquad 1\leq i\leq n.$

(i): If has only trivial idempotents (i.e.: 0 and ) then each $E_{ii}$ is a primitive idempotent of .
(ii): If is a local ring then each $E_{ii}$ is a local idempotent.
(iii): If is a division ring then each $E_{ii}$ is a division idempotent.

When $R=\mathbb{R}\oplus\mathbb{R}$ then (i) is not satisfied and consequently neither are (ii) and (iii). When $R=\mathbb{Z}$ then (i) is satisfied but not (ii) nor (iii). When $R=\mathbb{R}[[x]]$ - the formal power series ring over $\mathbb{R}$ - then (i) and (ii) are satisfied but not (iii). Finally when $R=\mathbb{R}$ 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 $\mathbb{Z}$ .

"idempotent classifications" is owned by Algeboy.

(view preamble)

Also defines:

division idempotent, local idempotent

Keywords:

primitive idempotent

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Cross-references: matrix rings, simple rings, artinian, Wedderburn-Artin theorems, consequence, formal power series, matrices, primitive, anti-idempotent, onto, projects, left invertible, Jacobson radical, conjectures, integers, integral domain, maximal ideal, ring, local ring, division ring, product, idempotent, unital ring

This is version 5 of idempotent classifications, born on 2007-03-07, modified 2007-03-08.
Object id is 9046, canonical name is IdempotentClassifications.
Accessed 1179 times total.

Classification:

AMS MSC:	20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)
	16U99 (Associative rings and algebras :: Conditions on elements :: Miscellaneous)

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