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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...

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    What is a maximal maximal ideal?2011-01-23

2 Answers 2