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. :)

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- Feb 21st 2006, 08:32 PMsuedenationAlgebra, 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. :) - Feb 22nd 2006, 02:46 PMThePerfectHackerQuote:

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.