Linked Questions
10 questions linked to/from Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition
3
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$A$ has only finitely many minimal prime ideals $\implies\ (0)$ is decomposable? [duplicate]
Let $A$ be a commutative ring with only finitely many minimal prime ideals.
Is the zero ideal $(0)$ decomposable?
[The converse implication is well known. Recall that an ideal is decomposable if it ...
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39
votes
4
answers
6k
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Commutative non Noetherian rings in which all maximal ideals are finitely generated
In commutative rings we have the following
Theorem. $R$ is Noetherian if and only if each prime ideal of $R$ is finitely generated.
From this Theorem I am looking for commutative rings $R$ in which ...
- 645
10
votes
3
answers
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Example of a finitely generated faithful torsion module over a commutative ring
Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$?
It's not difficult to show that there is no such ...
- 5,805
3
votes
1
answer
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An example of ideal that has no primary decomposition.
Give an example of a commutative ring with unit and an ideal that has no primary decomposition.
I think boolean Ring will be the right example, but I don't know how I must show that. So please help ...
- 2,881
21
votes
1
answer
1k
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Are finitely generated projective modules free over the total ring of fractions?
Let $Q(A)$ be the total ring of fractions of a commutative reduced non-noetherian ring $A$. Let $P$ be a finitely generated projective module over $Q(A)$ which is of constant rank (i.e. locally free ...
- 233
3
votes
3
answers
306
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Non-domain of Krull dimension zero
Let $F$ be a field, and $V$ be an $F$-vector space. Make $R=F⊕V$ a ring by putting $xy=0$ for $x,y\in V$. Is it true that the Krull dimension of $R$ is equal to zero?
If this is so, $R$ would be an ...
- 5,700
3
votes
1
answer
498
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Is it true that if some power of an ideal is primary, then the ideal itself is also primary?
Is it true that if some power of an ideal $I$ is primary, then $I$ itself is also a primary ideal?
I do not know whether the above statement is true or there is a counterexample. If one wants to ...
- 33
6
votes
2
answers
241
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Is there a non-artinian noetherian ring whose non-units are zero-divisors?
Is there a non-artinian noetherian ring whose non-units are zero-divisors?
Equivalent formulation:
Is there a noetherian ring of positive dimension whose non-units are zero-divisors?
[In this post, ...
- 19.4k
5
votes
1
answer
170
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An example of a finitely cogenerated ring having infinitely many maximal ideals
A commutative unital ring $R$ is called finitely cogenerated if an intersection of ideals of $R$ is zero, then a finite intersection of them is also zero. I am looking for a finitely cogenerated ...
- 51
1
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Non Classical Examples of Indecomposable Ideals
A classical example of a ring $R$ with an indecomposable ideal is the ring $C(X)$ of real valued continuous functions on $X$, where the $(0)$ ideal is not decomposable.
Does anyone know other examples ...
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