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

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$A$ 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
77 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
52 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
71 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
40 views

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
48 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
25 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
55 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
77 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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2answers
60 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
37 views

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
53 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
146 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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25 views

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
39 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
63 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
88 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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0answers
39 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
41 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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2answers
97 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 prime ideal $Q$ 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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1answer
47 views

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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1answer
56 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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0answers
34 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 $\mathcal{S}$ ...
3
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1answer
103 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 ...
4
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1answer
33 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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33 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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1answer
32 views

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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2answers
242 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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1answer
54 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
59 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 ...
2
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1answer
50 views

An easy question about fractional ideals…

Let $A$ be an integral domain and $K$ its field of fractions. If $M$ is a non-zero fractional ideal of $A$, then $$N=\{x \in K : xM \subseteq A\}$$ is also a fractional ideal of $A$. The proof I am ...
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1answer
32 views

Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$

Suppose $R$ is integral domain and $K$ is the fraction field of $R$. Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$.
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1answer
43 views

Proving $C\otimes_A\Omega^1_{A/R} \cong \Omega^1_{C/B}$

I am completely stuck on this so any help would be great. Let $R$ be a commutative ring and let $A$ and $B$ be $R$-algebras. Let $C:=A\otimes_RB$. Show that $C\otimes_A\Omega^1_{A/R} \cong ...
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2answers
49 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
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1answer
61 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
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37 views

Localization at a prime and direct limits

Let $R$ be a commutative ring with $1 \neq 0$ and let $P \subset R$ be a prime ideal. Apparently we have $$\varinjlim\limits_{f \in R \setminus P} R_f \cong R_P$$ where $R_f$ the the localization of ...
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1answer
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How do I compute the normalisation of $A=k[X,Y]/(Y^3 - X^5)$?

I'm trying to solve exercise 4.7 in Reid's UCA: "Find the normalisation of $A=k[X,Y]/(Y^3 - X^5)$." I can easily show $A$ is not normal: let $x$ and $y$ denote the images of $X$ and $Y$ in $A$. Thus ...
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1answer
69 views

If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties: $I_n$ is generated by a (finite) number of polynomials which ...
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1answer
67 views

a math-software that can compute analytic spread

I want to compute "analytic spread" . So I need a math-software that can compute it. can anyone help please? Here is the definition:
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1answer
55 views

Is a flat coherent sheaf over a connected noetherian scheme already a vector bundle?

Let $A$ be a connected noetherian ring (not necessarily irreducible), $M$ be a finitely presented flat $A$-module. Then $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for each $\mathfrak{p} ...
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2answers
58 views

Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
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1answer
42 views

Exercise of commutative algebra, rational functions.

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints. ...
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1answer
59 views

Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
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Ring with nested prime ideals [closed]

If $n>1$ is there a (commutative with identity) ring with Krull dimension $n$ and only $n+1$ prime ideals?
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1answer
35 views

A condition that an algebraic set is irreducible.

From the book by Kenji Ueno, Algebraic Geometry 1. From Algebraic Varieties to Schemes: "If an algebraic set $V(J)$ is reducible, it can be expressed as: $$(1.8)\quad V(J)= V(J_1)\cup V(J_2), \ ...
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0answers
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Computing homomorphisms between extensions of modules

Suppose we have two exact sequences of $R$-modules ($R$ is a commutative ring) $$0\rightarrow M_0\rightarrow F\rightarrow M_1\rightarrow0$$ $$0\rightarrow N_0\rightarrow G\rightarrow ...
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For a ring homomorphism, why does $f$ induces a homeomorphism from $SpecB$ onto the closed subset $V(\ker f)$ of $SpecA$.

Let $\varphi : A \rightarrow B$ be a ring homomorphism. Then we have a map of sets $Spec(\varphi):Spec(B) \rightarrow Spec(A)$ defined by $p \mapsto \varphi^{-1}(p)$ for every $p \in SpecB$. ...