# Doubt about proof the smallest subfield that contains a ring

Proposition. Let $$F$$ a field and $$R\ne\{0\}$$ a subring such that $$1_F\in R$$. We place $$F'=\bigg\{ab^{-1}\;|\;a\in R, b\in R\setminus{\{0\}}\bigg\},$$ the $$F'$$ is the smalles subfield of $$F$$ which contains $$R$$.

The I did not understand just one thing in the proof: because if $$1_F\in R$$, then $$R\subseteq F'$$. Would anyone be kind enough to explain it?

Thanks!

• The ring must be much more than just that: it must be an integral domain...And you also must define the operations in that $\;F'\;$ that you defined. – DonAntonio Nov 20 '18 at 17:54

Take any $$a\in R$$ and $$b=1_F$$. Then $$a=ab^{-1}\in F'$$. Hence $$R\subseteq F'$$.