# Thread: Ring theory, some questions...

1. ## Ring theory, some questions...

Hey everybody, thank you for reading.
I have an exam soon, and I have accumulated some questions which I haven't found a solution for... I'd appreciate it if I could get a push...

1)Given a ring in which every x in R satisfies x^3=x, I need to show it's commutative, and fail to see why. (note: this question was given a little after teaching the basic definitions and notions of "ideals".)
2) Given a set which is almost a ring, besides the fact that we're not given that a+b=b+a (abelian group under addition), that has a unit element, I need to show that the condition a+b=b+a must be satisfied anyhow.
'note: there's a hint here: expand (a+b)(1+1) in 2 ways. Haven't mannaged to do anything with that, however.

That's it for now :-)
Thanks!!!
Tomer.

2. What do you mean by "has a unit element"?

Maybe this helps.....
(a+b)(1+1)=(a+b)1+(a+b)1
and
(a+b)(1+1)=a(1+1)+b(1+1)

3. By "has a unit element" I mean there exists an element 1 in the set that satisfies 1*x=x*1=x for all x... That's not one of a Ring's axioms, and apperently it's existence implies that a+b=b+a (that's the Ring axiom I'm not given).

And, thanks for the tip, but I obviously tried this and other combinations and couldn't see where it leads me....

4. Consider (1+1)(a+b). We can distribute from the left:

(1+1)a + (1+1)b = a + a + b + b

We can also distribute it from the right:

1(a+b) + 1(a+b) = a + b + a + b.

Setting these two equal to each other and canceling the a on the left and the b on the right yields a + b = b + a.

5. Thanks