Does anyone know how to show the following:
Suppose that R is a ring with identity, and suppose that 1R = 0R. Show that
R = { 0R }.
Thx guys.
I understand it as,Originally Posted by suedenation
$\displaystyle 1R=0R$
But $\displaystyle 0R=\{0\}$ because $\displaystyle 0x=0,\forall x\in R$.
Thus,
$\displaystyle 1R=\{0\}$.
Thus,
But, $\displaystyle 1R=R$ because $\displaystyle 1x=x,\forall x\in R$.
Thus,
$\displaystyle R=\{0\}$.
Q.E.D.