Question 152
Let $(R,+, \cdot)$ be a ring with the property that, for every element $x \in R$, there exist two elements $e_{1}$ and $e_{2}$ of $R$ such that $e_{1}^{2}=e_{1}$, $e_{2}^{2}=e_{2}$ and $x=e_{1} e_{2}$. Show that:
(a) $1$ is the only invertible element of the ring $R$; and
(b) $x^{2}=x$, for every $x \in R$.
(a) $1$ is the only invertible element of the ring $R$; and
(b) $x^{2}=x$, for every $x \in R$.