Question

.....

Answer 1

Well, this is straightforward. Any idea how to begin?

Answer 2

Well, you begin by taking two arbitrary elements of R, and for simplicity, let's call them x and y. And you have to show that xy = yx.

Answer 3

try
\[xyxy=xxyy\] and multiply by the appropriate inverses left and right

Answer 4

Well
xyxy = x^2y^2
xyxy = xxyy

Answer 5

Just a thought, @satellite73
It might not be proper to think along the lines of inverses and more like left and right cancellation, as we don't know if multiplicative inverses exist...

Answer 6

Hang on, I'm thinking...

Answer 7

Now, if x equals zero, this whole thing would be trivial, right?
xy = 0y = 0 = y0 = yx
Same goes for y = 0
xy = x0 = 0 = 0x = yx

So let's assume that neither of them are zero.

Answer 8

Catch me so far?

Answer 9

xyxy = xxyy
Then
xyxy - xxyy = 0
Correct?

Answer 10

Well, this is a ring, so you can un-distribute the x, to the left, giving you
x ( yxy - xyy ) = 0

Answer 11

Well, since x is not zero, then
yxy - xyy = 0

Answer 12

hmmm... I'm stumped again :D

Answer 13

It seems to be a good, if somewhat messy way to prove it. I don't think I can top it, though.

Answer 14

I have to go to bed now, sorry :(