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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 ....
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