Unformatted text preview: a ∈ R is called a zero divisor if there are nonzero b and c such that ab = 0 and ca = 0. (The text only de±ned zero divisors for commutative rings.) (a) Prove that every nonzero nilpotent element of a ring is a zero divisor. Suppose R is a commutative ring and suppose a, b ∈ R satisfy a n = 0 and b k = 0. It can be shown that ( a − b ) n + k = 0. (You do NOT need to do this.) (b) Prove: If R is a commutative ring, then the nilpotent elements of R are an ideal. (c) It was shown in class that the only ideals in the ring M 2 ( R ) of 2 × 2 real matrices are the trivial ones { } and M 2 ( R ). Use this to show that “commutative” is necessary in (b). Hint : Look at a = p 0 1 0 0 P and b = p 1 P . END OF EXAM...
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 Spring '08
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 Math, Algebra, Ring, 5 pts, Ring theory, Commutative ring, nonzero nilpotent element, zero divisor

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