Let $R$ be a ring, $S\subset R$ a multiplicative subset, and let $M$ be a Noetherian $S^{-1}R$-module, then does there exist some Noetherian $R$-module such that $S^{-1}N\cong M$? What about if we only consider localizations of the form $R_f$? What if we also require $R$ to be Noetherian? If we let $N={_R M}$ then clearly […]

I’m considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the maximal ideal is exactly $(x)$. I also want to determine if $R$ is a Noetherian ring, but I have […]

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian $A$-algebras. The more examples the better. In other words, I’m asking a big list of examples.

I’m trying to prove that $\mathbb Z[\sqrt {-5}]$ is Noetherian. I already know that $\mathbb Z[X]$ is Noetherian and I’m trying to find a surjective map $$\varphi: \mathbb Z[X]\to \mathbb Z[\sqrt{-5}]$$ with $\ker\varphi=(X^2+5)$. If I could find this map I could prove that $\mathbb Z[\sqrt{-5}]\cong \mathbb Z[X]/(X^2+5)$ and then $\mathbb Z[\sqrt{-5}]$ is Noetherian. Thanks

Let $R$ be a commutative ring with unity such that all maximal ideals are of the form $(r)$ where $r\in R$ and $r^2=r$. I wish to show that $R$ is Noetherian. I know that if all prime (or primary) ideals in $R$ are finitely generated, then $R$ is Noetherian, so my plan was to show […]

If $R$ is noetherian, show that the polynomial ring of infinite variables $R[x_1,x_2,…]$ is coherent, i.e. every finitely generated ideal is finitely presented. I don’t really know how to get started. I tried to use that over a noetherian ring any finitely generated module is also finitely presented. But a finitely generated ideal over the […]

Let $R$ be a commutative ring with unity , $M$ be a finitely generated module over $R$ , let $N,P$ be submodules of $M$ such that $P\subseteq N \subseteq M$ and $M\cong P$ , then is it true that $M\cong N$ ? If not true , then what happens if we also assume that $M$ […]

Does there exist a Noetherian domain which is not a field , whose field of fractions is ( isomorphic with ) $\mathbb C$ ?

The problem is as follows: Let $R\subseteq S$ be an integral extension and $S$ a Noetherian ring. Then show that for each $\mathfrak p\in \operatorname{Spec}R$, there are only finitely many $P\in \operatorname{Spec}S$ such that $P\cap R = \mathfrak p$. Is $R$ also Noetherian? I am able to show the first part, but unable to show […]

Let $R\subset A\subset R[X]$ be commutative rings and suppose $A$ is Noetherian. Is $R$ Noetherian? I guess the answer is yes. Can we say from this relation that $A[X]=R[X]$? If yes, then by Hilbert Basis Theorem, $A[X]$ is Noetherian, hence $R[X]$ is Noetherian $\Longrightarrow R$ is Noetherian.

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