Given $R$ an integral domain (commutative ring with no zero divisors), and $\mathfrak P$ a prime ideal in $R$, is there a relation between the field of fractions of $R$ and the field of fractions of $R/\mathfrak P$?
It's trivial to see that whenever $\mathfrak P$ is also maximal, then $\text{Frac}(R/P)\cong R/\mathfrak P$, but in general it would be nice if thing worked like that:
1) There exists at least a maximal ideal containing $P$
2) There exists a maximal maximal ideal $\mathfrak M$ containing $P$
3) the field of fractions of $R/\mathfrak P$ is $R/\mathfrak M$
but I'm not able to prove or disprove this...