As a part of my studies in ring theory, I've encountered the concept of an idempotent element, i.e., an element $e$ such that $e^2=e$.
Are there some interesting examples of rings with idempotent elements?
Abstract Algebra – Examples of Rings with Idempotent Elements
abstract-algebraexamples-counterexamplesring-theory
Best Answer
As you realize, all rings have the idempotents $0$ and $1$, so the question is whether they have any others.
If a commutative ring has a non-trivial idempotent, then it is isomorphic to a product of two non-trivial rings. The same is true for a non-commutative ring, as long as the idempotent lies in its centre.
If $n$ is a positive integer which is not a prime power then $\mathbb{Z}/n\mathbb{Z}$ has nontrivial idempotents.
Matrix rings tend to have lots of idempotents, but not usually in their centres.
A group algebra $KG$ for a finite group $G$ over a characteristic zero field $K$ has the central idempotent $|G|^{-1}\sum_{g\in G}g$ and usually others.