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

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Can anyone help me understand an application of Nakayama lemma?

In the Wikipedia there is an application of Nakayama lemma: In the special case of a finitely generated module $M$ over a local ring $R$ with maximal ideal $m$, the quotient $M/mM$ is a vector ...
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1answer
24 views

Prove that some local noetherian integral domain is a field

A local noetherian integral domain $A$ is a field if the unique maximal ideal $m$ satisfies $m^n = m^{n+1}$ for some $n\in N$ I think it should be related to Nakayama lemma, but cannot figure it ...
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3answers
56 views

Commutative artinian ring is noetherian

Suppose R is a commutative Artinian ring then R is Noetherian. I am aware of the proof which uses the idea of filtration. But I would like to prove this fact without that idea but haven't got far ...
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1answer
73 views

A problem with tensor products

Let $K$ be a field, $R=K[x^2,x^3]$, $S=K[x]$, and consider $S$ as an $R$-module. Given $f: S \to R \oplus R$ so that $f:p \mapsto (x^3p,-x^2p)$, prove that $f\otimes 1: S \otimes_R S \to (R\oplus R) ...
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1answer
29 views

Understanding the proof (via primary decomposition) the “ideal factorization theorem” in Dedekind domains

I am trying to understand the outline of the strategy for proving (via primary decomposition) that every non-zero ideal of a Dedekind domain can be expressed uniquely (up to the order of the factors) ...
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53 views

Is this ideal a prime ideal?

Let $k$ be a field and $k[x_1,x_2,x_3,x_4]$ a polynomial ring in four variables over $k$. How can we show that the ideal $(x_3^3-x_2^2x_4, x_4^3-x_1^2x_3, x_3x_4-x_1x_2, x_2x_4^2-x_1x_3^2)$ is prime? ...
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1answer
52 views

Confusion about the definition of localization

For example, let $\mathbb Z$ be the ring and $S = \mathbb Z - 2\mathbb Z$. Then the quotient ring should be: $S^{-1}A = \{a/s: a\in \mathbb Z \text{ and }s \in S\}$, which is formed by the equivalence ...
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2answers
35 views

Image of the map induced on spectra

Apologize in advance if this is a bit trivial but I am stuck on the following: Prove that for $\varphi : R \to S$ a map between commutative rings, the prime $\mathfrak{p}$ is in the image of the ...
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0answers
24 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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2answers
59 views

Counterexample for non-prime ideal [closed]

Let $A$ be a commutative ring and $P$ be a prime ideal. Then $S = A - P$ is a multiplicatively closed subset. My question is if it is true that $S$ is a multiplicatively closed subset, then $A-S$ ...
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47 views

Prime ideals in $A$ and prime ideals in $S^{-1}A$

Let $A$ be a ring and $S$ be a multiplicative closed subset. Then there is a 1 to 1 correspondence between the prime ideals in $A$ (intersect $S$ is empty) and prime ideals in $S^{-1}A$. My question ...
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3answers
52 views

$k[t^{a_1},t^{a_2},t^{a_3}]$ in the form $k[x,y,z]/(…)…(…)$

I want to write $k[t^6,t^7,t^{15}]$ in the form $k[x,y,z]/(...)...(...)$; but I even don't know how to start. is there in general a way that one can write $k[t^{a_1},t^{a_2},t^{a_3}]$ in the ...
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1answer
35 views

construction of $S^{-1}A$

If $A$ is a ring and $S$ a multiplicative set, how does the elements of $S^{-1}A$ look like? In my book one introduces the equivalence relation $\sim$ on $A \times S$ as follows: $(a,s) \sim (b,t) ...
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4answers
64 views

Counterexample in Dedekind domains

Let $K$ be a number field and $\mathcal O_K$ the ring of algebraic integers in $K$. If $\mathfrak p$ is a prime ideal, then $\mathcal O_K/\mathfrak p$ is finite field. My question is: Finding a ...
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2answers
40 views

$A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring

Question: Suppose $A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring. I have no idea how to construct the unique maximal ideal.
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1answer
71 views

Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$ and $\sqrt{IJ}\neq\sqrt{I}\sqrt{J}$ [closed]

Let $\Bbb k$ be a field and $I$, $J$ be ideals in $\Bbb k [x_1,x_2,\ldots,x_n]$. Show that: $(i)\; \sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$, $(ii)\; \sqrt{IJ}\neq\sqrt{I}\sqrt{J}$.
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1answer
28 views

Associated non-minimal prime ideal

I am trying to find an example of a noetherian local ring with an associated prime of height greater or equal 1. That is, I want a noetherian local ring $R$, together with an associated prime $p = ...
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0answers
65 views

How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
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70 views

Ambiguity in the definition of unmixed ideal

Compare the definitions: Page 136 Matsumura, Commutative ring theory: A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal. ...
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1answer
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Is an ideal prime when its complex extension is prime?

Let $I = \langle f_1,\dots,f_k\rangle$ be an ideal in $\mathbb R[x_1,\dots,x_n]$. The same $f_i$ generate an ideal $\widetilde I$ in $\mathbb C[x_1,\dots,x_n]$. When $\widetilde I$ is prime in ...
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0answers
33 views

Flatness on the affine line for a coherent sheaf

Let $A:=\mathbb{C}[t], M$ a finitely generated $A$ module. Denote by $m_\alpha$ the maximal ideal generated by $t-\alpha$ for $\alpha \in \mathbb{C}$, $S_\alpha$ the multiplicative set which is the ...
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1answer
38 views

Extension of R-linear derivation to localization

Given an $R$-algebra $A$, a multiplicative subset $S \subset A$, and a $R$-linear derivation $D: A \rightarrow M$, where $M$ is an $S^{-1}A$-module, $D$ can be uniquely extended to a $R$-linear ...
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1answer
24 views

$\overline{V(I)-V(J)}=V(\bigcup_{n=1}^{\infty}I\colon J^n)$

Is it true that $\overline{V(I)-V(J)}=V\left(\bigcup_{n=1}^\infty I\colon J^n\right)$? If not, is it true for noetherian rings?
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1answer
44 views

Show that $Rad(I)$ is a prime ideal

The ring $R$ is commutative with unit. An ideal $I$ is called primary, if it stands the following: If $ab \in I$ then $a \in I$ or $b^n \in I$, for a natural number $n$. Show that if $I$ is a ...
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1answer
66 views

Does any (noetherian) integral domain have a “UFD closure”?

Let $R$ be a (possibly noetherian if that helps) commutative unital integral domain. Does there exist a UFD $\overline{R}$ such that $R$ embeds in $\overline{R}$ (via some map $\psi$) and such that ...
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3answers
94 views

Show that quotient rings are not isomorphic

I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic. This is my attempt at a solution: For ...
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2answers
45 views

Irreducible components of affine variety

Fix some algebraically closed field $k$ and let $X$ be the affine variety given by the ideal $I=(z^2-xy,xz-z)$, how can I describe the irreducible components of $I$? I know that there is a bijection ...
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2answers
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Localization Question: $\frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)}$

Questions: $\rm\color{#c00}{(1)}$ Is the $[\Longrightarrow]$ implication of $$ \frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)} $$ obvious? ...
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2answers
41 views

Prime ideals in an arbitrary direct product of rings

By ring I mean commutative unital ring. The prime ideal structure of a finite direct product of rings is well known: For $\prod_{i=1}^n R_i$, it is of the form $\prod_{i=1}^n P_i$ where only one ...
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0answers
132 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
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Computing other valuations of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are indeterminant over $k$, I know valuations of $F$, the field of fractions of the ring $A=\dfrac{k[x,y]}{I}$ where $I$ is an ideal of ...
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1answer
35 views

Question on Generic Freeness, ref. [Matsumura, page 185]

I am sure this must have been answered somewhere but I can't find them, so I shall try my luck here. Let $A$ be a Noetherian integral domain and $M$ a finitely generated $A$-module. Then there ...
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1answer
47 views

Discrete Valuation Rings - Atiyah & MacDonald

The following is claimed (without much proof) during the the proof of Prop 9.2 in Atiyah & MacDonald. Saurabh commented below giving the proof that was probably intended by A&M (thank you!). I ...
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1answer
70 views

When is the quotient ring of a multivariable polynomial ring over a field an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field an integral domain? I am actually trying to show that a monomial ideal is prime by showing the corresponding quotient ...
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36 views

Henselization of the ring of polynomials

I am trying to understand example of Henselization from wiki. http://en.wikipedia.org/wiki/Henselian_ring#Henselization It says that Henselization of the ring of polynomials localized at point $(0, ...
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1answer
40 views

Generic flatness on modules

I am looking for a stronger notion of generic flatness. Let $A$ be a Noetherian ring, $M$ a finitely generated module over $A$. Suppose there exists a maximal ideal $m$ of $A$ such that $M_m$ (the ...
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1answer
66 views

In a Noetherian integral domain, a principal prime ideal can't have proper non-zero prime ideals

Let $R$ be an integral domain and Noetherian. Let $P \subset R$ be a non zero prime ideal. Prove that if $P$ is principal then there is no $Q$ prime ideal such that $0 \subsetneq Q \subsetneq P$. ...
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How to construct a unique valuation for $k\left(T_{i}\right)_{i\in\mathbb{N}}$ in $\mathbb{Z}^{\left(\mathbb{N}\right)}$?

Let $k$ be a field and $\left(T_{n}\right)_{n\in\mathbb{N}}$ indeterminates over $k$. Let $K=k\left(T_{n}\right)_{n\in\mathbb{N}}$ and $\varGamma:=\mathbb{Z}^{\left(\mathbb{N}\right)}$ the abelian ...
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Artin local ring [duplicate]

I have studied "structure theorem for Artin rings" which states "An Artin ring $A$ is unique a finite direct product of Artin local rings". Let $A$ be Artin ring. By Chinese remainder theorem, $A ...
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53 views

Is $A \rightarrow S^{-1} A$ epi?

this question must be the most stupid I have ever asked. If $A$ is a commutative ring and $S$ a multiplicative subset the usual inclusion induces a homeomorphism onto the image $\text{Spec} ...
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30 views

Intersection theorems for a certain type of subsets of integers modulo $N$

I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ ...
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1answer
59 views

Two elements in a non-integral domain which are not associates but generate the same ideal

Let $\mathbb{K}$ be a field. Let $R$ be the quotient ring $\mathbb{K}[x,y]/(xy^{2})$. Let $\bar{x}$ be the class of $x$ in $R$ (i.o.w. $\bar{x}=x+(xy^{2}))$. Prove that $\bar{x}$ and ...
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28 views

Irreducible radical ideals are prime

Assume $R$ is a commutative ring and $I$ is a nonzero proper ideal of $R$ satisfying: $(1)$ If $I_1$ and $I_2$ are ideals such that $I = I_1 \cap I_2$, then $I = I_1$ or $I = I_2$; $(2)$ If $a^n ...
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30 views

Length of chain of prime ideals in polynomial ring

Let $B=A[x_1,...,x_n]$ be a ring of polynomials over the ring $A$, $P$ be a prime ideal in $A$. Suppose that we have the chain $Q_0\subset Q_1\subset ... \subset Q_k$ of strictly embedded prime ideals ...
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The non flat module $\mathbb{Z}/m$.

A general result states that an $R$-module $M$ is flat if and only if $I\otimes_R M \simeq IM$ for all ideals $I\subset R$. However, there is something I don't understand. Let $R = \mathbb{Z}$, and ...
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230 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
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examples of interpreting schemes (Eisenbud)

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative ...
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What is the kernel of $I/I^2 \to \Omega_{\mathbb P^{n}/k} \otimes \mathcal O_X$?

Recall that if $X \subset \mathbb P^n$ is a smooth projective variety, we have the conormal sequence of locally free sheaves on $X$ (here $I$ is the ideal sheaf of $X$): $$ I/I^2 \xrightarrow{\delta} ...
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1answer
48 views

Localization of Rings: Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$

Let $R$ be a ring, $f \in R$, and $X$ a variable. Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$. I am a beginner in algebra and I am reading a textbook in commutative algebra. What I do not ...
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2answers
57 views

Two ways to localize a ring using a prime ideal

I was reading the part about localization of the Introduction to Commutative Algebra of Atiyah-MacDonald and I have a question I was not able to solve. Let $R$ be a commutative ring with unit $1$ and ...