Examples for when the quotient ring is necessarily

Janiyah Hays

Janiyah Hays

Answered question

2022-04-03

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 m. Let F:=Frac(R) be the field of fraction of R.

Answer & Explanation

etsahalen5tt

etsahalen5tt

Beginner2022-04-04Added 8 answers

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

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