Definition

For a ring , an idempotent is an element such that . An idempotent is called central if it is in the center of .

Central idempotents as ideal and ring

Let be a central idempotent. Let . Then is a 2-sided ideal.

Proof: Let ,

is also a ring.

Proof:

This means that the identity element in is .

If is a central idempotent, then so is since

Thus we decompose using these two elements, for any ,

this gives

as rings.

Note: This process can be generalized using orthogonal families of central idempotents.