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If every maximal ideal is finitely generated is the ring Noetherian? [duplicate]

$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
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Rings in which every maximal ideal is finitely generated [duplicate]

Suppose that $R$ is a commutative ring with unity in which every maximal ideal is finitely generated. Then is $R$ Noetherian?
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Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition

Recently I encountered a problem (the first exercise from chapter four of Atiyah & McDonald's Introduction to Commutative Algebra) stating that if $\mathfrak{a}$ is a decomposable ideal of $A$ (a ...
644 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
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Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
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The height of a principal prime ideal

A formal consequence of Krull's principal ideal theorem is the following: If $A$ is a Noetherian ring, and $I$ is an ideal generated by $r$ elements, then any prime ideal which is minimal among those ...
623 views

Finitely generated ideal containing non finitely generated ideal

I've been thinking about the following Rotman's exercise, and just can't find an answer: Give an example of a commutative ring $R$ containing proper ideals $I\subsetneq J\subsetneq R$ with $J$ ...
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Is there a domain which is not UFD but has a maximal principal ideal?

Maximal ideals in $\Bbb Z[\sqrt{-5}]$, which are not UFD, are not principal. I wonder, however, a maximal ideal could be principal. Is there known example? Also, I wonder the existence of UFD that has ...
Example of a finitely generated ideal $I$ with $R/I$ is Noetherian and $R$ not Noetherian. [closed]
Can somebody give me example of ring $R$ such that $R/I$ is Noetherian but $R$ is not Noetherian ring? $I$ is finitely generated ideal of $R$. Also please search example such that $I$ is not ...