Question

Show that if R is ring with unit element and 1=0,then R=(0)

Answer 1

Let \(r\in R\). It can be proved that \(0\cdot r=0\) (usually available in the text book). Isnce \(1=0\) then
\[r=1\cdot r=0\cdot r=0\]
Therefore \(R=\{0\}\).

Answer 2

Hello

Answer 3

Can you explain more please

Answer 4

which part that you don't understand?

Answer 5

you divide 0/Cdot

Answer 6

that is just multiplication in the ring

Answer 7

what about if I said if 0=1 , a0=a that means 0=1

Answer 8

yes, if 0=1 the statement a0=a is a correct statement

Answer 9

that means I can answer it like this

Answer 10

like what? what did you show there?

Answer 11

if I said 0a=a

Answer 12

so what if you show that 0a=a?

Answer 13

i ask you if that correct if i said 0a=a is that mean 0=1?

Answer 14

no, you said that if 1=0 is it correct that a0=0 and 1=0? So I said yes

Answer 15

so my question is that means i have to suppose that a=1

Answer 16

no. In the beginning we don't suppose anything about a. We just take any \(a\in R\). Next we use the property of the unity element 1. So \(a=1a\). But it is given that 1=0, so we can replace teh 1 with 0. Therefore we have
\[
a=1a=0a
\]
But 0a=0. Therefore a=0

Answer 17

thank you