# Algebra, Ring theory

• February 21st 2006, 09:32 PM
suedenation
Algebra, Ring theory
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. :)
• February 22nd 2006, 03:46 PM
ThePerfectHacker
Quote:

Originally Posted by suedenation
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,
$1R=0R$
But $0R=\{0\}$ because $0x=0,\forall x\in R$.
Thus,
$1R=\{0\}$.
Thus,
But, $1R=R$ because $1x=x,\forall x\in R$.
Thus,
$R=\{0\}$.
Q.E.D.