# Ring and fields

(a) If R is a field, show that R itself is a field of fractions for R.

(b) Show that Q is a field of fractions for Z and for 2Z.

please see attached pdf.

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#### Solution Preview

If R is a commutative ring and S is a subset of R - {0} containing no zero divisors, let Q := R_S denote the ring of fractions of R with respect to S.

Then, as we have seen, we can embed R into Q; and every element of Q may be written in the form

ab^{-1}

for some a in R, and b in S, since every element of S becomes a unit in Q (with inverse b^{-1}, i.e. b^{-1} = e/be, for any e in S, where b = be/e in Q).

If F is a field, then the field of fractions Q of F consists of elements of the form ab^{-1} with a in F and b nonzero in F. ...

#### Solution Summary

This provides examples of two proofs regarding fields of fractions.