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Rings question

Let R be a ring in which x^2=x for all x€R where x^2 of course denotes x*x
a) prove that x+x=0 for all x €R
b) prove that R is commutative

For a) I'm thinking: assume for a contradiction that x+x is not equal to 0 for all x €R.
Since x^2=x then x=1,0. Choose x=0 so we have 0+0=0 reaching a contradiction.

Is that right? Feels too simple.

But even if x+x=0 for all x€R, if x is 1 then it doesn't even work.

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Original post by cooldudeman
...


Usual caveat about rings.

However your logic is in error here

" x+x=0 for all x €R"

Could be written as ∀x∈R:x+x=0\forall x\in \mathbb{R}: x+x=0

What's the negation of that?
Original post by ghostwalker
Usual caveat about rings.

However your logic is in error here

" x+x=0 for all x €R"

Could be written as ∀x∈R:x+x=0\forall x\in \mathbb{R}: x+x=0

What's the negation of that?


What do u mean by negation?
Also I have never heard of caveat of rings...

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Original post by cooldudeman
What do u mean by negation?



You went for a proof by contradiction. That starts by assuming the negation of what you're trying to prove and showing that this leads to a contradiction, from which you conclude your negation was false, and the original statement true.

PS: A proof by contradiction isn't likely to get you anywhere.



Also I have never heard of caveat of rings...


"caveat" - dictionary or Google it.
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Avatar for davros
davros
16
Original post by cooldudeman
What do u mean by negation?
Also I have never heard of caveat of rings...

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caveat is just Latin for "let him beware" - he meant be aware of his usual comment about being rusty on rings :smile:
Original post by davros
caveat is just Latin for "let him beware" - he meant be aware of his usual comment about being rusty on rings :smile:


Ohhh lol. I think ghostwalker is a she though

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