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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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...