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Finite ring proofs

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Please see the attached pdf. I need a detailed, rigorous proof of this with explanation of the steps so I can learn.

Let R be a finite ring.
a. Prove that there are positive integers m and n with m>n such that x^m for ever x E R.
b. Give a direct proof (i.e. without appealing to part c) that if R is an integral domain, then it is a field
c. Suppose that R has identity, prove that if x E R is not a zero divisor, then it is a unit

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Solution Summary

This solution provides a series of proofs regarding a finite ring and positive integers, integral domain, and zero divisor.

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(a) Since R is finite, assume |R| = n, then for any non-zero element x in R, we have x^n = x.
Now let m = n^2 > n, then x^m = x^(n^2) = (x^n)^n = x^n
For x = 0, we always have x^m = x^n = 0.
Therefore, we find m>n, such that for all x in R, x^m = x^n.
(b) For each nonzero element a in R, ...

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