This preview shows pages 1–2. 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: Math 4124 Monday, April 11 April 11, Ungraded Homework Exercise 7.1.5 on page 230. Decide which of the following (a) (f) are subrings of Q : (a) the set of all rational numbers with odd denominators (when written in lowest terms) (b) the set of all rational numbers with even denominators (when written in lowest in terms) (c) the set of nonnegative rational numbers (d) the set of squares of rational numbers (e) the set of all rational numbers with odd numerators (when written in lowest in terms) (f) the set of all rational numbers with even numerators (when written in lowest terms) Let R be the described set, so we need to decide whether R is a subring of Q . (a) Yes. (i) 0 R because 0 = / 1. (ii) Suppose a / r , c / s R where r , s are odd. Then obviously a / r R , and a / r + b / s = ( as + br ) / rs . This is in R because rs is odd and when we reduce to lowest terms, the denominator will still be odd. (iii) Suppose a / r , c / s R . Then ( a / r )( c / s ) = ( ac ) / ( rs ) ; since rs is odd and the de nominator will remain odd when reducing to lowest terms, it follows that R is closed under multiplication. (b) No, because R is not closed under addition. For example 1 / 2 , 1 / 2 R , but 1 / 2 + 1 / 2 = 1 / 1 / R ....
View
Full
Document
This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.
 Spring '08
 Staff
 Math, Algebra

Click to edit the document details