Is R a Commutative Ring if xx=x for all x in R?

[boombaby]

Homework Statement

Let R be a ring. If any x in R satisfies xx=x, prove that R is a commutative ring.
[1 is not necessarily in R in the definition of ring according to this particular book]

Homework Equations

The Attempt at a Solution

I made some attempts but failed. I have ab=aabb=abab. However, a(ab-ba)b=0 does not implies ab-ba=0, since cancellation law does not necessarily hold...
Any hint? I fail to draw any conclusion more than just ab=aabb=abab...

 

[cristo]

I'm a bit rusty on this sort of thing, but haven't you shown that R is commutative? If you've got aabb=abab, then can't you just multiply on the left with a^-1 and on the right with b^-1, giving ab=ba?Edit: Probably ignore that: rings don't necessarily have multiplicative inverses, do they?

 

[morphism]

Edit: Probably ignore that: rings don't necessarily have multiplicative inverses, do they?

Nope.

Instead I'd first show that 2=0 in this ring. Then I'd think about how one would run into something that looks like ab-ba (=ab+ba) -- and especially something that involves squares.

 

[boombaby]

1, I make no progress...
2, since 1 is not required to be in the ring according to the definition of ring in this book ...perhaps, 2=0 is not necessarily be true, as it is proved (perhaps?) by 2=1+1 and is therefore in R. and 2=2*1=2*(2-1)=2*2-2*1=2-2=0
Okay, now I find something weird...is it proper to write xx+x=x(x+1) if 1 is not in R? I'm getting more confused I'll check the book later.
3, perhaps it is ok for me to assume 1 is in R. We will see whether and how 1 is used in a proof that works.
4, I make no progress...

 

[boombaby]

Anyone?

 

[morphism]

We don't need 1 to be in R.

(1) By considering (x+y)^2, prove that xy+yx=0.
(2) Then prove that xy+xy=0.

Now here's a challenge for you: can you find a (nonzero) ring R without 1 such that x^2=x for all x in R?

 

[boombaby]

Here's what I thought:
R, {3k|k is integers} mod 6, is such a commutative ring without '1' in it. And we have 3*3=3, 0*0=0.
however, it seems that 3 really acts as 1 in this ring, since 3*3=3 and 3*0=0. So it is not a ring without 1...but I've no idea how to get a ring without a real 1 at the moment...

BTW, I have one thing to clarify...If 1 is required in the definition of ring, should the subring inherit the same 1 from its original ring? I guess no. If yes, it would cause something strange...but I'm not quite sure about it right now..

back to the topic..could you give me hint to find a ring without 1 such that xx=x?

Thanks for all the help...

 

[morphism]

Sorry for the late reply. I forgot about this thread!

BTW, I have one thing to clarify...If 1 is required in the definition of ring, should the subring inherit the same 1 from its original ring? I guess no. If yes, it would cause something strange...but I'm not quite sure about it right now..

I guess we're discussing this in your other thread.

back to the topic..could you give me hint to find a ring without 1 such that xx=x?

Take the power set of some set S. Define A+B to be symmetric difference of A and B, and define AB to be the intersection of A and B. Verify that S is a ring and that AA=A for all A in S. S does have a '1' though - what is it?

Can you use this example to come up with a ring without 1 such that xx=x?

Or how about taking an infinite direct sum of copies of Z_2?

 

[boombaby]

I've been thinking about The Boolean ring defined on S (since it is one of the few rings with the property xx=x that I know). However, if S is well defined, it seems that S is always the 1...
I'll be thinking on this carefully later, and will raise another thread if I encounter problems. Thanks very much:)