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Well, I'll give part A a try, even though I don't know what a unitary ring
is, and aren't sure I remember what a ring is. I remember there were rings,
fields, and groups.
Maybe as I go I'll remember stuff !
We're given that for every x in R,
x^3 = x
2x=0
Writing these longer,
x*x*x = x
x+x=0
So, I must assume that rings have two operators associated with them, "*"
and "+".
Then there's that 0. I'll assume that the element 0 is some existing element
in R (must it be unique ?) such that for every x,
x+0=x
Anyway, we want to show that given those 2 statements, that for every x and
y in R, it is true that
x*y = y*x
But hmm. Maybe we're supposed to prove
x+y = y+x
Or maybe we can prove *both* ???
Well, let's start with
x*y
What can we do with it ? We can apply the cube premiss:
x*x*x*y
What else can we do ? We could apply it to y as well:
x*x*x*y*y*y
Maybe we should apply it to the middle (replace x*y with its cube):
x*x*x*y*x*y*x*y*y*y
Actually, we were sloppy in that step, since we really started with
(x*x*x)*(y*y*y)
and in order to cube the middle, we must be sure we're allowed to regroup as
(x*x)*(x*y)*(y*y)
This, of course, presupposes associativity, that is, for every x and y, and
z,
(x*y)*z = x*(y*z)
Is this a given quality of rings ?
Oh well, maybe I'll wait for some answers to my elementary questions before
proceeding.
Thanks.
/Eric
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