I need an example of a ring consisting of 2 by 2 matrices where $a^3=a$ with $a$ belonging to this ring. If someone can list the elements I would be satisfied.
What I'm trying to get at it is conceptualize why a ring $R$ is always commmuative when $a^3=a$. I know of one such example and that is the factor ring $\mathbb{Z}/3\mathbb{Z}.$ Does anyone know how to prove this statement mathematically as well as giving me an example of a ring of 2 by 2 matrices?