idempotent classifications

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

Lemma 1.

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

Proof.

Suppose $e\\in R$ with $e\eq 0$ and $e^{2}=e=1e$ . Then by cancellation $e=1$ .∎

The integers are an integral domain which is not a division ring and theyserve as a counter-example to many conjectures about idempotents of generalrings as we will explore below. However, the first important result is toshow the hierarchy of idempotents.

Theorem 2.

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

Proof.

Let $\\mathfrak{m}$ be the unique maximal ideal of $R$ . Then $\\mathfrak{m}$ is the Jacobson radical of $R$ . Now suppose $e\\in\\mathfrak{m}$ is anidempotent. Then $1-e$ must be left invertible (following theelement characterization of Jacobson radicals (http://planetmath.org/JacobsonRadical)). So there exists some $u\\in R$ such that $1=u(1-e)$ . However, this produces

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 $1$ .Thus if $e\\in R$ , $e\eq 0$ is an idempotent then it projects to anidempotent of $R/\\mathfrak{m}$ and as $e\otin\\mathfrak{m}$ it follows $e$ projects onto $1$ so that $e=1+z$ for some $z\\in\\mathfrak{m}$ . As $e^{2}=e$ we find $0=z+z^{2}$ (often called an anti-idempotent). Once againas $z\\in\\mathfrak{m}$ we know there exists a $u\\in R$ such that $1=u(1+z)$ and $z=u(1+z)z=u(z+z^{2})=0$ so indeed $e=1$ .∎

Corollary 3.

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

Example 4.

Let $R$ be a unital ring. Then in $M_{n}(R)$ the standard idempotentsare the matrices

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

(i)
If $R$ has only trivial idempotents (i.e.: $0$ and $1$ ) then each $E_{ii}$ is a primitive idempotent of $M_{n}(R)$ .
(ii)
If $R$ is a local ring then each $E_{ii}$ is a local idempotent.
(iii)
If $R$ 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 consequentlyneither 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 seriesring 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 simplerings 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 $\\mathbb{Z}$ .