Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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indecomposable summand

Let $M$ be an $R$ module. Is this true for start a proof that we say "Let $S$ denote the indecomposable summands of $M$"? In fact, I want to know whether any module over a Dedekind domain (or a ...
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If $\{f_1,…,f_n\}$ generate $R$ then does $\{f_1^N,…,f_n^N\}$ [duplicate]

Let $R$ be a commutative ring such that $\{f_1,...,f_n\}\subseteq R$ generates $R$. Does this imply that for all integers $N>0$ that $\{f_1^N,...,f_n^N\}$ generates $R$? I would have guessed not, ...
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Exercise 5.5.F. on Ravi Vakil's Notes related to associated points [duplicate]

Let $A$ be a Noetherian ring and $M$ a finitely generated $A$ module. In Ravi Vakil's notes he first states that the associated points of $M$ satisfy the following: (A) The associated points of $M$ ...
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Irreducible closed subsets of a scheme corresponds to points

I have posted an answer here for the case of an affine scheme, but I got stuck when I tried to generalize the argument to schemes. My thoughts Consider a point $p$ in the scheme, its closure in the ...
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what can be said about $spec(R_m)$, where $R_m$ is localization of $R$ at maximal ideal $m$

I've seen how if $p$ is a prime ideal of $R$ and $R_p$ is the localization of $R$ at $P$, then $P_p$ is the unique maximal ideal of $R_p$, but what if we had a maximal ideal $m$ of $R$, then $R_m$ ...
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Trouble understanding Eisenbud Exercise 2.19a

I'm working through the "Commutative algebra with a view toward algebraic geometry" book and stumbled onto an exercise I'm struggling to answer. Let $R$ be a ring and let $M$ be an $R$-module. ...
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Calculating Spec of the localization $R_P$

I am studying a first course in commutative algebra and I'm currently working through some exercises on calculating $Spec(R_P)$, where $R_P = R[(R\backslash P)^{-1}]$ is the localization of $R$ at a ...
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How do I find the spectrum of a ring?

What is $Spec R$ where $R$ is the integers modulo $6$? More generally, what are the techniques to find the spectrum of any commutative ring? (I would also be interested in the non-commutative case but ...
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Is a product of two Noetherian schemes over Spec $\mathbb Z$ a Noetherian scheme?

In Hartshorne's proof of Proposition 6.6 in Chapter 2, he says that if $X$ being Noetherian implies $X\times\mathbf A^1$ is "clearly" Noetherian. I assume this is because $X$ can be covered by affine ...
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Ring with maximal ideal not containing a specific expression

Main question : May there exist an integral domain $R$, with fraction field $K$, that fulfills the following condition: there exists $x\in K$, $x\not \in R$ and a maximal ideal $\frak m$ of $R[x]$, ...
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Example that inverse limit is not exact

Its known that "inverse limit is not exact". Matsumura in his book Commutative Ring Theory, page 272, gives an example for this. I can not understand how he proves that inverse limit of $Z$ is zero. ...
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Converse to Chinese Remainder Theorem

So as seen on this question Converse of the Chinese Remainder Theorem, we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, ...
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If $I$ is a homogeneous ideal of $A$ contained in $A_+$, then $\sqrt{I} = \bigcap\limits_{I\subset P\in\text{Proj }A} P$?

EDIT: This is from an exercise of Vakil's Foundations of Algebraic Geometry. 4.5.H: Suppose $I$ is any homogeneous ideal of $S$ contained in $S_+$, and if $f$ is a homogeneous element of ...
if $A$ is a module,then the family fin($A$) of all the finitely generated submodules of $A$ is a directed set and direct limit of$M_i$ is isomorphic to$A$. for prove this needed to define to injection ...