ProblemSet9 - it has nontrivial right ideals.) (b) Let R be...

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

View Full Document Right Arrow Icon
Homework 9 Due: Monday, April 12, 2010 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. Show that there is an infinite number of solutions to x 2 = - 1 in the ring of quaternions { α 0 + α 1 i + α 2 j + α 3 k | α i R } . 2. Show that ±² a b - b a ³ ´ ´ ´ a,b R µ is a field. 3. If R is a finite integral domain, show that R is a field. (Do not assume that all integral domains contain an identity. Although some books add that assumption to the definition, we did not.) 4. (a) Show that the ring of 2 × 2 matrices over the reals has nontrivial left ideals. (Similarly,
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: it has nontrivial right ideals.) (b) Let R be the ring of 2 2 matrices over the reals and suppose that I is an ideal of R . Show that I = (0) or I = R . 5. Let R = Z [ i ], the ring of Gaussian integers . Starting with R , construct a eld having 49 elements. 6. Find all the units (invertible elements) in the ring Z 24 7. Let R be a ring in which x 4 = x for every x R . Prove that R is commutative. 8. If a,b Z such that either 3 6 | a or 3 6 | b (or both) show that 3 6 | ( a 2 + b 2 )...
View Full Document

This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.

Ask a homework question - tutors are online