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Apr 14, 2021 at 18:56 comment added user237522 Please, I meant, same question and in addition to the conditions of the question we further assume that $S=R[w]$, for some $w \in S$.
Apr 14, 2021 at 18:10 comment added Will Chen @user237522 I don't understand your question
Apr 14, 2021 at 13:23 comment added user237522 Thank you very much! Interesting. Please, what if $S$ is a simple extension of $R$, namely, $S=R[w]$, for some $w \in S$? (perhaps your answer also answers this; I do not know..).
Apr 14, 2021 at 13:16 vote accept user237522
Apr 14, 2021 at 3:37 history edited Will Chen CC BY-SA 4.0
added 80 characters in body
Apr 14, 2021 at 3:28 history answered Will Chen CC BY-SA 4.0