(b) Prove that if x is an idempotent in an integral domain (that is, $x^2 = x$), then x = 0 or x = 1. (c) Suppose s is an idempotent in a ring S with unity and s ? 0 and s ? 1. Find zero divisors t and u in S with tu = 0.
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To prove that if x is an idempotent in an integral domain, then x = 0 or x = 1, we can use the fact that an integral domain has no zero divisors. Assume x is an idempotent in an integral domain. This means that x^2 = x. Now, let's consider the case where x ≠ 0. Show more…
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