oh sorry. Thanks François Brunault. This question is true in case R be PID.

I Think that, fixing F or R is the same question. Because R is a arbitrary domain.

This question is true in case F be PID.

I want to known this problem true or false for all domain F.

thanhs Ralph very much, [ Let F be a domain and let R≤F be a subring such that F is a free R-module of finite rank n. Is there an R-basis {e1,...,en} of F such that at least one of the basis elements is a unit in F ? ] ok. I want to known this problem true or false.

Thanks Ralph [Are e1,...,en are given in advance or do you want to know if a basis e1,...,en can be choosen such that at least one of them is an unit in F ?] I want to know if a basis e1,...,en can be choosen such that at least one of them is an unit in F ?]

As an example consider R=Z and F=Z[i]=R⋅1⊕Ri.?? we can chose a base 1,i of Z[i] the 1 i unit !

I mean, F is a domain, R≤F a subring such that F is a free R-module , Can we conclude that some ei(1≤i≤n) is a unit of F with basis e1,...,en.? (can We chose a base of F?)