s09_mthsc851_hw10 - MTHSC 851 (Abstract Algebra) Dr....

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Unformatted text preview: MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 10 Due Friday April 24th, 2009 (1) Let R and S be commutative rings, and let f : R S be a ring homomorphism. (a) If f is surjective and I is an ideal of R , show that f ( I ) is an ideal of S . (b) Show that part (a) is not true in general when f is not surjective. (c) Show that if f is surjective and R is a field, then S is a field as well. (2) Let p be a fixed prime number, and consider the ring R = { a b : a, b Z , ( a, b ) = 1 , p- b } with the usual operations of addition and multiplication of rational numbers. (a) Determine the group of units of R . (b) Prove that the principal ideal ( p ) = pR is a maximal ideal of R , and in fact the only maximal ideal of R . (3) Suppose R is an integral domain and P R is a prime ideal. (a) Show that both P and R \ P are multiplicative semigroups. (b) If S = R \ P show that U ( R S ) = R S \ R S P . Conclude that R S P is the unique maximal ideal in R S ....
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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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