We should define what it means for a ring element to be idempotent and prove some basic properties about them.
We should also link this to decomposition of $R$-modules. if $e$ is an idempotent element of $R$ then an $R$-module $M$ can be decomposed as $M \cong eM \oplus (1-e)M$. This will be important for an eventual proof of the Wedderburn-Artin theorem.
We should define what it means for a ring element to be idempotent and prove some basic properties about them.
We should also link this to decomposition of $R$-modules. if $e$ is an idempotent element of $R$ then an $R$-module $M$ can be decomposed as $M \cong eM \oplus (1-e)M$. This will be important for an eventual proof of the Wedderburn-Artin theorem.