Integral Domain, r^2 = r proof that r = 0 or 1

[RJLiberator]

Homework Statement

Let r be an element of an integral domain R such that r^2 = r. Show that either r = 0_R or 1_R

Homework Equations

integral domain means no zero divisors.

The Attempt at a Solution

This is fundamental as 0 and 1 solve r^2 = r and are the only solutions.

However, I'm not really sure what I can play with to show this fact.

We have r*r = r
There are no zero divisors so no such thing as r*s = 0.

If we start this proof off by assuming r is not 1_R or 0_R then maybe we could get somewhere, but It doesn't feel promising.

There has to be a way to use the fact that r^2 = r.

(r^2)*r = r^2 ?
Then we are left with r*r = r, that's no good.

Is there any trick that I am neglecting? I feel this is just a simple multiplication of two things and boom we have r(r^2) = 0 or 1 or something.

How about r^2 - r = r-r
then we have
r(r-1)=0
r is either 0 or 1.

BOOM?

 

[RJLiberator]

When the problem talks about such things as 0_R and 1_R is it safe to assume that they just mean 0_R = 0 and 1_R =1 ?

For instance, my proof has the following:

r^2 = r
r^2 - r = r - r
r(1-r) = 0
Therefore r = 0 or r = 1

But is that a complete proof in consideration of 0_R and 1_R ?

 

[Samy_A]

Homework Statement

Let r be an element of an integral domain R such that r^2 = r. Show that either r = 0_R or 1_R

Homework Equations

integral domain means no zero divisors.

The Attempt at a Solution

This is fundamental as 0 and 1 solve r^2 = r and are the only solutions.

However, I'm not really sure what I can play with to show this fact.

We have r*r = r
There are no zero divisors so no such thing as r*s = 0.

If we start this proof off by assuming r is not 1_R or 0_R then maybe we could get somewhere, but It doesn't feel promising.

There has to be a way to use the fact that r^2 = r.

(r^2)*r = r^2 ?
Then we are left with r*r = r, that's no good.

Is there any trick that I am neglecting? I feel this is just a simple multiplication of two things and boom we have r(r^2) = 0 or 1 or something.

How about r^2 - r = r-r
then we have
r(r-1)=0
r is either 0 or 1.

BOOM?

The last one is correct. BOOM!

 

[RJLiberator]

Samy_A do you have a moment to check out my reply? I am wondering on the terminology of this problem. :)

 

[Samy_A]

When the problem talks about such things as 0_R and 1_R is it safe to assume that they just mean 0_R = 0 and 1_R =1 ?

For instance, my proof has the following:

r^2 = r
r^2 - r = r - r
r(1-r) = 0
Therefore r = 0 or r = 1

But is that a complete proof in consideration of 0_R and 1_R ?

Yes. 0_R is the identity element for addition, 1_R is the identity element for multiplication. It is common to simply write them as 0 and 1, as long as it is clear what ring we are talking about.

 

[HallsofIvy]

"$0_R$" and "$1_R$" specifically mean "the additive identity" and the "multiplicative identity" of the ring, R, respectively. I believe SamyA is saying that it is alright to drop the "R" if it is clear what ring you mean. You cannot assume that they are the numbers 0 and 1 if that was what you meant.