The identity of a domain and the identity of its field of fractions Let D be a domain and let F be its field of fractions. If e is the identity of the domain D and e' is the identity of its field of fractions F, can we say that e=e'?

Would you help me, please?
Thank you in advance.
This has nothing to do with domains and field of fractions: 

Lemma. If $G$ and $H$ are groups such that $H \subset G$, then $e_H=e_G$. 

Proof. $e_H=e_H * e_G=e_G.$

The answer to your question follows from setting $H=D$ and $G=F$.
This has nothing to do with domains and field of fractions: 

Lemma. If $G$ and $H$ are groups such that $H \subset G$, then $e_H=e_G$. 

Proof. $e_H=e_H * e_G=e_G.$

The answer to your question follows from setting $H=D$ and $G=F$.