Doubt about proof the smallest subfield that contains a ring

Question

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!

Answer 1

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