What are some examples of non-noetherian rings $R$ for which $\dim R[T]=\dim R+1$ holds?
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In general, if $\dim R=n$ we have $n+1\le\dim R[T]\le2n+1$. For $n=0$ this gives $\dim R[T]=\dim R+1$. Now choose your favorite ring of dimension zero which is not noetherian.
user26857
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If you choose $R$ a countable direct product of (copies of) $\mathbb F_2$, then notice that $R$ is coherent. – user26857 Nov 22 '15 at 16:36
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where can I find a proof of the first inequality? Thank you! – Eleonora Mar 01 '20 at 11:30
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@vevvostar This is Theorem 38 in Kaplansky, Commutative Rings. – user26857 Mar 01 '20 at 22:40