Linked Questions

2
votes
1answer
650 views

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?
1
vote
0answers
37 views

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?
11
votes
1answer
2k views

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 ...
12
votes
2answers
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 ...
20
votes
1answer
782 views

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 ...
4
votes
1answer
1k views

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 ...
8
votes
1answer
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 $J$ ...
2
votes
1answer
202 views

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 ...
0
votes
0answers
174 views

An ideal is maximal iff it is generated by an irreducible elemnt

It is well known that in a PID an ideal is maximal iff it is generated by an irreducible element. My query is 'is the result true only in a PID?'. My question is what will happen if the PID condition ...
1
vote
1answer
80 views

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 ...