s09_mthsc851_hw09

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

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

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the 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