This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: HOMEWORK 3 DUE: Fri., Oct. 14 NAME: DIRECTIONS: Turn in your homework as SINGLESIDED typed or handwritten pages. STAPLE your homework together. Do not use paper clips, folds, etc. STAPLE this page to the front of your homework. Be sure to write your name on your homework. Show all work, clearly and in order . You will lose point 0.5 points for each instruction not followed. Questions Points Score 1 1 2 1 3 1 4 1 5 2 6 1 7 2 8 1 Total 10 1 HOMEWORK 3 DUE: Fri., Oct. 14 NAME: Problem 1: (1 point) (a) (0.5 point) Show that { ( a, b ) } is an additive inverse for { ( a, b ) } . Proof: Consider { ( a, b ) } + { ( a, b ) } = { ( ab + b ( a ) , b 2 ) } = { (0 , b 2 ) } = { (0 , 1) } , which is the additive identity in Q . Q.E.D. (b) (0.5 point) Prove the distributive law for Q . Proof: Let { ( a, b ) } , { ( c, d ) } , and { ( e, g ) } Q . We want to show { ( a, b ) } [ { ( a, b ) } + { ( e, g ) } ] = { ( a, b ) } { ( c, d ) } + { ( a, b ) } { ( e, f ) } . A little algebra shows that L.H.S. = { ( a, b ) } { ( cf + de, df ) } = { ( acf + ade, bdf ) } , R.H.S = { ( ac, bd ) } + { ( ae, bf ) } = { ( acbf + bdae, b 2 df ) } = { ( b, b ) } { ( acf + dae, bdf ) } , but { ( b, b ) } = { (1 , 1) } which is the multiplicative identity in Q so Q.E.D. Problem 2: (1 points) Let R be a ring and R a nonempty subset of R . Show that R is a subring iff, for any a , b R , we have a b , ab R ....
View
Full
Document
This note was uploaded on 10/18/2011 for the course MATH 19900 taught by Professor Schmidt during the Fall '07 term at UChicago.
 Fall '07
 SCHMIDT

Click to edit the document details