# Rationals with an odd denominator form a ring

Consider Q, the set of all rational numbers. For qâ??Q, q=m/n for m, n â?? Z. Suppose this is written in reduced form so gcd(m,n)=1. Let R â?? Q consist of all rational numbers for which the denominator of the reduced form is odd. Show that this set forms a ring under the usual operations of addition and multiplication.

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#### Solution Preview

We need to check that:

1) R is closed under the operations of addition and multiplication:

Suppose, p=a/b and q=c/d belong to R. Then their product pq = (ac)/(bd), and the denominator (bd) is an odd number, since 2 is prime. Suppose, the ...

#### Solution Summary

We show that rational numbers with an odd denominator form a ring.

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