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Questions tagged [commutative-algebra]

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

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

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Is the inverse limit an additive functor in the category of $R$-mod?

Is the inverse limit an additive functor in the categories of $R$-mod? Equivalently, if we have two inverse systems $\{A_i\}$ and $\{B_i\}$ of $R$-modules sharing the same index set, do we have $$ \lim_{\leftarrow} A_i \oplus B_i = \lim_{\leftarrow}…
k99731
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Is the completion of regular ring regular?

This is probably an elementary problem but I wasn't able to figure it out. Given a regular ring $A$ and an ideal $I$ which you can assume $A/I$ is also regular. Is the completion ring $\hat{A}$ necessarily regular? This is true if $A$ is local but…
user127776
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Why is this map an isomorphism?

In Altman-Kleinman "Introduction to Grothendieck Duality Theory", page 105, there is the following Lemma: I don't understand why by uniqueness, I can conclude $w \circ w'=id$. Any help?
Jotabeta
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$\mathfrak{p}$ a prime ideal, $\varphi: A\rightarrow S^{-1}A$, then $\varphi(\varphi^{-1}(\mathfrak{p}))$

Let $A$ be a commutative ring, $S$ a multiplicatively closed subset. Let $S^{-1}A$ denote the localization. I want to show that for $\mathfrak{p}$ a prime ideal, $\varphi: A\rightarrow S^{-1}A$, the ideal generated by…
Mark Murray
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A question about Noetherian one-dimensional domains.

If $R$ is a one-dimensional Noetherian domains, not Dedekind. It is true that each localization with a maximal ideal is a DVR, since the localization is a one dimensional Noetherian local domain. My question is: is this reasoning correct, even if…
Rick88
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Integral extensions and number of generators

here's a doubt which arised from a previous question: Suppose $R$ is a ring and $S \subseteq R$ is a subring. Moreover suppose $R$ is integral over $S$ and $R$ is finitely generated as an $S$-module. In a previous post Matt E showed that we always…
user6495
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Notation problem in polynomial rings. What is $\mathfrak{p}R[x]$?

I have seen a lot the notation "$\mathfrak{p}R[x]$", where $\mathfrak{p}$ is a prime ideal and $R[x]$ is a polynomial ring with coefficients in $R$. My first question is: What is $\mathfrak{p}R[x]$ exactly? I think that it is not the same as…
Gerardo Vargas Flores
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Why is true this part of Krull's height Theorem

I am stuck trying to prove this part of Krull's height Theorem. I checked this proof and, in page 18, it says that if a prime ideal $\mathfrak{p}$ contains $f_1, g_2,\dots,g_n$ this implies that $f_1, f_2^{m},\dots,f_n^{m}\in\mathfrak{p}$ and hence…
Gerardo Vargas Flores
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Arithmetic rank and height

I've been looking for an example of an ideal whose height and Arithmetic rank is different. I think that it must differ just in 1, but i can't imagine an example. $\mathbb{ara}(I)=\mathbb{ht}(I)+1$
user801020
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definition of finitely generated $A$ algebra, $A$ a ring

Let $A$ be a ring. I'm a bit confused with definition of $A$ algebras and I would appreciate any clarification. Let $A \subset B$ rings and suppose $B$ is a finitely generated $A$ algebra. Does this mean there exist $x_1, .., x_n \in B$ for some $n$…
Johnny T.
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Definition of a graded ring

In Matsumura's commutative ring theory book he says that if R is ring graded by an abelian semigroup then $R_0$ is a subring. In particular he states that $1\in R_0$. But this seems wrong if we just take the product of rings. For example…
alex pacun
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A question about integral exension of domains.

If $A, B$ are two domains and $Q(A), Q(B)$ their fields of fractions, it is true that, if $A \subseteq B$ is an integral extension then $\bar{A}=\bar{B}$? Where with $\bar{A}$ we denote the integral closure of $A$ in its field of fractions. I think…
Rick88
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Question about normal module.

Let $k$ be a field and let $\phi: S\to R=S/I$ be $k$-algebras. Then given a homomorphism $\varphi: I\to R$, it's said that it induces a homomorphism $\psi:I/I^2\to R$ since $\varphi$ kills $I^2$. I am not sure about this: if $\varphi(x)$ is…
Yuyi Zhang
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Does every prime ideal of a ring contain a minimal prime?

Let $R$ be a ring and $\mathfrak p \subset R$ a prime ideal. I think that it is easy to show that $\mathfrak p$ contains a minimal prime ideal of $R$, by the following argument: pass to the localization $\varphi : R \to R_\mathfrak p$, fetch a…
isekaijin
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How to show that an $\mathbb{Z}[\sqrt{-5}]$-module is not free?

I do not know how to start with the following exercise: Let $R := \mathbb{Z}[\sqrt{-5}]$. Show that the left $R$-module $M:=\big\langle 2, 1+\sqrt{-5} \big\rangle$ (so the $R$-module generated by $2$ and $1+\sqrt{-5}$) is not free. I do not see…
3nondatur
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