Where does (x^2y - x^2yx^2)^2 come from? Can I just make up anything that will help match up x^3 = x, or is that expression something I should already know?
It is only possible when ring satisfies the property x^3=x for all x in a ring. But in Z8 , 2^3=8=0 not equal to 2.Hence Z8 does not satisfy the property.
If x^2=0 for some x in a ring R and x^3=x for every x in R, then x=x^3=x.x^2=x.0=0.
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at 9:00, we have used the property a + (-a) =0, as R is a group with respect to +. (Not the cancellation laws.)
Can we prove it using one-one onto, polynomial ring?
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At 19:00 why cant we replace x^2 by x-x^2 directly as x^2 can be anything.
Where does (x^2y - x^2yx^2)^2 come from?
Can I just make up anything that will help match up x^3 = x, or is that expression something I should already know?
It is just an expression which has been used to solve x^3=x for the commutativeness of the ring and nothing else.
What will happen if (x²y-x²yx²) become divisor of zero ?
Can we then write (x²y-x²yx²)²=0 => (x²y-x²yx²)=0 ???
For example, in (Z8, +,.)
4²=0 but 4 nonzero
It is only possible when ring satisfies the property x^3=x for all x in a ring.
But in Z8 , 2^3=8=0 not equal to 2.Hence Z8 does not satisfy the property.
If x^2=0 for some x in a ring R and x^3=x for every x in R, then x=x^3=x.x^2=x.0=0.