Let $R$ be a commutative ring with 1, and $e^2=e$ be an idempotent element of $R$. Recall that an irreducible idempotent is an idempotent $a$ for which $Ra $ is a simple $R$-module.
Question: Is the following true: An idempotent element $e$ of $R$ is irreducible if and only if the ring $Re $ is a field?
Yes, suppose that $e$ is an irreducible idempotent. We are going to show that $Re$ is a field with neutral $e$. Let $x$ a non zero element of $R$ The submodule of $Re$ generated by $xe$ is $Re$. So there exists $y$ such that $yxe=e=yexe$. We deduce that the inverse of $xe$ is $ye$.