s09_mthsc851_hw10

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.

Ask a homework question - tutors are online