idempotent classifications

An a unital ring R, an idempotentMathworldPlanetmathPlanetmath eR is called a division idempotent if eRe={ere:rR}, with the productMathworldPlanetmathPlanetmathPlanetmath of R, forms a division ring. If instead eRe is a local ringMathworldPlanetmath – here this means a ring with a unique maximal idealMathworldPlanetmath 𝔪 where eRe/𝔪 a division ring – then e is called a local idempotent.

Lemma 1.

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


Suppose eR with e0 and e2=e=1e. Then by cancellation e=1. ∎

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.

Theorem 2.

Every local ring R has only trivial idempotents 0 and 1.


Let 𝔪 be the unique maximal ideal of R. Then 𝔪 is the Jacobson radicalMathworldPlanetmath of R. Now suppose e𝔪 is an idempotent. Then 1-e must be left invertible (following the element characterization of Jacobson radicals ( So there exists some uR such that 1=u(1-e). However, this produces


Thus every non-trivial idempotent eR lies outside 𝔪. As R/𝔪 is a division ring, the only idempotents are 0 and 1. Thus if eR, e0 is an idempotent then it projects to an idempotent of R/𝔪 and as e𝔪 it follows e projects onto 1 so that e=1+z for some z𝔪. As e2=e we find 0=z+z2 (often called an anti-idempotent). Once again as z𝔪 we know there exists a uR such that 1=u(1+z) and z=u(1+z)z=u(z+z2)=0 so indeed e=1. ∎

Corollary 3.

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

Example 4.

Let R be a unital ring. Then in Mn(R) the standard idempotents are the matrices

  1. (i)

    If R has only trivial idempotents (i.e.: 0 and 1) then each Eii is a primitive idempotent of Mn(R).

  2. (ii)

    If R is a local ring then each Eii is a local idempotent.

  3. (iii)

    If R is a division ring then each Eii is a division idempotent.

When R=RR then (i) is not satisfied and consequently neither are (ii) and (iii). When R=Z then (i) is satisfied but not (ii) nor (iii). When R=R[[x]] – the formal power series ring over R – then (i) and (ii) are satisfied but not (iii). Finally when R=R then all three are satisfied.

A consequence of the Wedderburn-Artin theorems classifies all ArtinianPlanetmathPlanetmath simple ringsMathworldPlanetmath as matrix rings over a division ring. Thus the primitive idempotents of an Artinian ring are all local idempotents. Without the Artinian assumptionPlanetmathPlanetmath this may fail as we have already seen with .

Title idempotent classifications
Canonical name IdempotentClassifications
Date of creation 2013-03-22 16:48:43
Last modified on 2013-03-22 16:48:43
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 9
Author Algeboy (12884)
Entry type Definition
Classification msc 16U99
Classification msc 20M99
Defines division idempotent
Defines local idempotent