s09_mthsc851_hw09 - MTHSC 851 (Abstract Algebra) Dr....

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MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 9 Due Tuesday April 14th, 2009 (1) Give an example of a ring with exactly 851 ideals. (2) If F is a field, show that M n ( F ) is a simple ring. (3) Let R be a ring with unity and x R any non-unit. Use Zorn’s lemma to prove that x is contained in a maximal ideal. (4) A local ring is a commutative ring with identity which has a unique maximal ideal. Prove that R is local if and only if the non-units of R form an ideal. (5) Let R be a finite ring. (a) Prove that there are positive integers m and n with m > n such that x m = x n for every x R . ( Hint : If | R | = n , then consider the ring S = R ×··· R , with n factors.) (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 R is not a zero divisor, then it is a unit. (6) (a) An element
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This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.

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