Math787_HW3

# Math787_HW3 - ab = 1 then ba must be also = 1 3 Let A be a...

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Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework III: Ring Theory 1. Let A be a not-necessarily commutative integral domain with 1. Suppose that a, b±A are such that ab = 1. Show that then ba = 1 and hence that a and b are units in A . 2. Let A be a ring and a, b±A so that ab = 1, but ba 6 = 1. Show that A has inﬁnitely many nilpotent elements and hence that if A is a ﬁnite ring and
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Unformatted text preview: ab = 1, then ba must be also = 1. 3. Let A be a ring that is not an integral domain. Show that the ideal I = { f±A [ x ] | f (1) = 0 } is never a prime ideal. 4. Let R be a ring with 1 and with | R | ﬁnite. Show that then every element in R is either a unit or is a zero-divisor. 5. Find all solutions of x 2-8 x + 5 = 0 in Z / 10 Z . 1...
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