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## Main Question or Discussion Point

May there exist an integral domain [itex]R[/itex], with fraction field [itex]K[/itex], that fulfills the following condition:

there exists [itex]x\in K[/itex], [itex]x\not \in R[/itex] and a maximal ideal [itex] \frak m[/itex] of [itex]R{[}x{]}[/itex], such that [itex] \frak m[/itex] does not contain [itex]x-a[/itex] for any [itex]a\in R[/itex] ?

Motivation : I am trying to prove a difficult result. A way to obtain it would be to show that if [itex] \varphi[/itex] is an epimorphism of an integral domain [itex]R[/itex] into a field [itex]F[/itex], then the residual field of every place [itex] \tilde \varphi[/itex] extending [itex] \varphi[/itex] to the fraction field of [itex]R[/itex], with finite values into an algebraic closure of [itex]F[/itex], is equal to [itex]F[/itex]. I have some doubts that such a miracle does occur; but this problem is not available in the literature.

Now, if the answer of the asked question is negative, then we are done, taking the restriction of [itex] \tilde\varphi[/itex] to [itex]R{[}x{]}[/itex] in the (allegedly) absurd supposition that such an extension of [itex] \varphi[/itex] exist.

there exists [itex]x\in K[/itex], [itex]x\not \in R[/itex] and a maximal ideal [itex] \frak m[/itex] of [itex]R{[}x{]}[/itex], such that [itex] \frak m[/itex] does not contain [itex]x-a[/itex] for any [itex]a\in R[/itex] ?

Motivation : I am trying to prove a difficult result. A way to obtain it would be to show that if [itex] \varphi[/itex] is an epimorphism of an integral domain [itex]R[/itex] into a field [itex]F[/itex], then the residual field of every place [itex] \tilde \varphi[/itex] extending [itex] \varphi[/itex] to the fraction field of [itex]R[/itex], with finite values into an algebraic closure of [itex]F[/itex], is equal to [itex]F[/itex]. I have some doubts that such a miracle does occur; but this problem is not available in the literature.

Now, if the answer of the asked question is negative, then we are done, taking the restriction of [itex] \tilde\varphi[/itex] to [itex]R{[}x{]}[/itex] in the (allegedly) absurd supposition that such an extension of [itex] \varphi[/itex] exist.