Examples for when the quotient ring is necessarily / not necessarily an extension of the residue field.Let R be an integral domain with unique maximal ideal&nbsp;m. Let&nbsp;F:=Frac(R)&nbsp;be the field of fraction of R.

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Examples for when the quotient ring is necessarily / not necessarily an extension of the residue field.Let R be an integral domain with unique maximal ideal&amp;nbsp;m. Let&amp;nbsp;F:=Frac(R)&amp;nbsp;be the field of fraction of R.

etsahalen5tt · Accepted Answer

Step 1I can at least explain the situation you linked to.Let R be a finitely generated algebra over an algebraically closed field k. Let&amp;nbsp;m&amp;nbsp;be a maximal ideal of R, and let&amp;nbsp;B=RmNow the quotient&amp;nbsp;BmB&amp;nbsp;is a field which is also a finitely generated k-algebra, so by Zariski&#039;s Lemma&amp;nbsp;BmB&amp;nbsp;is a finite extension of k. But now since k is algebraically closed, this forces&amp;nbsp;BmB&amp;nbsp;to just be k.So,&amp;nbsp;BmB&amp;nbsp;embeds in B, and thus also in&amp;nbsp;Frac(B)

Examples for when the quotient ring is necessarily

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