Linked Questions

3 votes
0 answers
286 views

$A$ 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 ...
Pierre-Yves Gaillard's user avatar
40 votes
4 answers
6k views

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 ...
Tomi Albertini's user avatar
10 votes
3 answers
2k views

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 ...
Censi LI's user avatar
  • 5,965
4 votes
1 answer
2k views

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 ...
kpax's user avatar
  • 2,941
21 votes
1 answer
1k views

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 ...
manoj's user avatar
  • 233
3 votes
3 answers
310 views

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 ...
karparvar's user avatar
  • 5,782
3 votes
1 answer
540 views

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 ...
user318830's user avatar
6 votes
2 answers
286 views

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, ...
Pierre-Yves Gaillard's user avatar
1 vote
0 answers
302 views

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 ...
Sabino Di Trani's user avatar
5 votes
1 answer
177 views

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 ...
Narnia's user avatar
  • 51