1. ## subfield proof

If R is a domain (integral domain), prove that there is no subfield K of Frac(R) such that

$\displaystyle R\subseteq K \subset Frac(R)$

Note: in the book the part that K is a subset of Frac(R) they used some funny notation, that basically looks like subset or equal to, but the line that means equal to was crossed out.... I just assumed that they mean the subset, which doesn't equal K.... but I'm not sure

2. Originally Posted by ux0
If R is a domain (integral domain), prove that there is no subfield K of Frac(R) such that

$\displaystyle R\subseteq K \subset Frac(R)$

Note: in the book the part that K is a subset of Frac(R) they used some funny notation, that basically looks like subset or equal to, but the line that means equal to was crossed out.... I just assumed that they mean the subset, which doesn't equal K.... but I'm not sure
if $\displaystyle r,s \in R, \ s \neq 0,$ then, since K is a field and it contains R, we have $\displaystyle s^{-1} \in K.$ thus $\displaystyle rs^{-1} \in K.$ but we know that Frac(R) is exactly the set of $\displaystyle rs^{-1}$ with $\displaystyle r,s \in R, s \neq 0.$ so K = Frac(R).