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

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Atiyah and MacDonald Theorem 9.5

$K$ is an algebraic number field, $A$ its ring of integers. Theorem 5.17 shows that $A\subseteq\sum\mathbb{Z}v_j$ with $v_j\in K$. Theorem 9.5 then concludes that $A$ is a f.g. $\mathbb{Z}$-module. I ...
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135 views

Infinite intersection of prime ideals

Let $A$ be a commutative ring with identity. Let $p_{i}, i\in I$ and $p$ be prime ideals in $A$, where the index set $I$ is infinite. If we have $$ p\supset \bigcap_{I}p_{i} $$ Do we still have $p\...
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158 views

“Closure” and “neighborhoods” in Spec(A)

While trying to work through the sequence of problems in Atiyah-Macdonald's first chapter regarding the prime spectrum of a ring, I've run across a small point of confusion. Namely: In the point ...
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85 views

Module is zero if localization at associated primes is zero

Let $A$ be a Noetherian ring and $M$ an $A$-module. I want to show that $M=0$ if $M_P = 0$ for each $P \in \text{Ass}(M)$. Here is my attempt at a solution: Assume for a contradiction that $M \not=...
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167 views

Depth of a module over local ring and vanishing of Ext functor

I'm studying depth of $A$-modules, where $A$ is a noetherian ring, in Matsumura's Commutative Algebra text and I'm experiencing some trouble understanding the proof of a basic result. I think all of ...
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1answer
73 views

Calculate the radical of ideals

Let $k$ be an algebraically closed field and consider $A=k[x,y,z]$. I am supposed to calculate $\text{rad}(x,y)= \{ f \in k[x,y,z] : f^n \in (x,y)$ $\text{for some n} \}$, $\text{rad}(x,z)$ and $\text{...
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145 views

Krull dimension of the quotient by a single element

Let $(R,m)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module of dimension $d$. The Krull dimension of $M$ is defined to be the Krull dimension of $R/\operatorname{ann}(M)$. ...
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95 views

Matsumura, CRT, Exercise 6.3

The questions states: Let $A$ be a Noetherian ring and $x\in A$ be an element which is neither a unit nor a zero-divisor; prove $\operatorname{Ass}_A(A/xA)=\operatorname{Ass}_A(A/x^nA)$ for each $...
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63 views

Definition of free module

i) Let $ M$ be a free $R$-module. By definition $ M = R \oplus R \oplus\cdots\oplus R$ . Can anyone could explain me why $ M = Rx_1 \oplus\cdots\oplus Rx_n$ where $x_1,\ldots,x_n$ elements of $M$. My ...
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115 views

If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents

Let $R$ be a commutative ring with identity, $P$ be a prime ideal in $R$ and define $$X := \lbrace t \in P \mid t^2=t \rbrace. $$ Also let $J$ denote the smallest ideal of $R$ that contains $X$. ...
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1answer
107 views

Find shortest primary decomposition.

Let $A=k[x,y,z]$ and let $T_1=(x,y)$, $T_2=(x,z)$. Define $I=T_1T_2$ and calculate the shortest primary decomposition of $I$. I dont know where to start and I am looking for hints, how should I think ...
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4answers
168 views

Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.

This is Atiyah-Macdonald Exercise 2.2 Exercise: Let $A$ be a ring, $\mathfrak a$ an ideal, $M$ an $A$-module. Show that $(A/\mathfrak a) \otimes_A M$ is isomorphic to $M/\mathfrak aM$. [Tensor the ...
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1answer
91 views

Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me a ...
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2answers
72 views

When a prime ideal is restricted to a basic open subset of projective space, is it still prime?

Suppose $I\subset k[x_0,\ldots,x_n]$ is a prime ideal. Now restricted on the basic open subset $\mathbb{P}^n_{x_i}$ of $\mathbb{P}^n$, is $I$ still prime? Note: 1. Here $\mathbb{P}^n_{x_i}$ is ...
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$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded $S$-...
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1answer
77 views

Restriction and extension of scalars between flat algebras and their completion over a DVR and ideals.

So, in a proof I am currently reading I have stumbled upon the following. Let $R$ be a discrete valuation ring, $\hat{R}$ its completion and $t$ a uniformizing parameter for $R.$ Let $A$ be a flat $R$...
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2answers
61 views

pictorial illustration of simplicial complexes

Consider the following two complexes (Bruns&Herzog p.215): By just looking at the complex on the left, i am not sure how to read its faces. Surely its vertices are $v_1,v_2,v_3,v_4,v_5$. The ...
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273 views

Invertible matrices in commutative rings

Let $A$ be a square matrix over a commutative ring $R$. Then $A$ has a left inverse iff it is invertible. Does there exist a elementary proof of this fact? (i.e. without using the determinant!)
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1answer
80 views

local PID that is not a field is a DVR

I would be very happy if someone would check my proof of the fact that a local PID that is not a field is a DVR: Let $A$ be a local PID that is not a field. Since irreducibles generate maximal ideals ...
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86 views

Question on rank function.

In a previous question I asked about the fiber $M(P)=M_P / PM_P$ where $M$ is an $A$-module and $P$ a prime ideal of $A$. Later I introduced the rank function $$rk_M : \text{Spec} A \to \mathbb{N} \...
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396 views

Is every prime element of a commutative ring “veryprime”?

Let $R$ denote a commutative ring. Define a function $$\| : R \times R \rightarrow \mathbb{N} \cup \{\infty\}$$ such that $a \| b$ is the number of times $a$ divides $b$ (and include $0$ in $\mathbb{...
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210 views

What is the algebraic tangent cone really?

Let $A$ be a (commutative unital) ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $B = A / \mathfrak{a}$. Then we have a descending filtration $$\cdots \subseteq \mathfrak{a}^3 \subseteq \...
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1answer
82 views

$IJ$ is the set of nilpotent elements

Let $R$ be a commutative ring with identity which is Noetherian. Let $V(A)$ denote the set of all prime ideals of $R$ containing the ideal $A$. Suppose that $V(0) = V(I) \cup V(J)$ and $V(I) \cap V(J) ...
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1answer
60 views

what inequalities can one have between $depth\ R$ and $depth\ M$?

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. what inequalities can one have between $depth\ R$ and $depth\ M$? Obviously there are ...
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2answers
86 views

David Eisenbud, Hilbert theorem

I just started reading D. Eisenbud Commutative algebra with a view towards algebraic geometry and I wonder about a theorem on page 42: If $M$ is a finitely generated graded module over $k[x_1,...,x_r]...
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1answer
111 views

Fiber as vector space over residue field.

Let $A$ be a commutative ring with identity and let $M$ be an $A$-module. The fiber of $M$ at $P \in \text{Spec}A$ is the module $M(P):=M_P / PM_P$, which is a vector space over the residue field $A(P)...
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1answer
87 views

Integral closure of a DVR in finite extension of fraction field

Let $(K,|\cdot|)$ be a complete valued field and let $L$ be a field extension with $[L:K]<\infty$. Let $\mathcal{O}_K$ be the valuation ring in $K$ and let $\mathcal{O}_L$ be the integral closure ...
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1answer
60 views

Notation for the number of times one element divides another.

Let $R$ denote a commutative ring with unity. Consider elements $a,b \in R$. Is there an accepted notation (like $a \| b$ or some such) for the number of times that $a$ divides $b$? Explicitly, we can ...
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171 views

A question on Mumford's drawing of $\text{Spec}\,\mathbb{Z}[x]$

This might seem like a really silly question, but what are those weird curves connecting $(x^2 + 1)$ and $(5, x+2)$ in Mumford's picture of $\text{Spec}\,\mathbb{Z}[x]$?
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104 views

Factorization of Artinian ring by its nilradical

Theorem. Let $R$ be a commutative Artinian ring with 1 over a field $k$ and $\mathfrak n$ be its nil radical. If char $k \ne 2$ then $R/\mathfrak n$ is isomorphic to a direct sum of fields. My proof. $...
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1answer
137 views

Direct-Sum Decomposition of an Artinian module

Let $R$ be a commutative Noetherian ring. Suppose $M$ is a finitely-generated non-zero Artinian $R$-module. Question: How can we prove that there are maximal ideals $m_1 , m_2 , \ldots , m_n$ such ...
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64 views

Units in a ring of fractions

Let $R$ be a UFD and $D \subseteq R$ multiplicative set. What are the units in $D^{-1}R$? I assume the answer should be $D^{-1}R^{\times}$, but I get stuck: If $a/b$ is a unit, then there exists $...
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1answer
45 views

example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$

What is an example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$? I've tried constructing one, but all my attempts have ...
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58 views

Annihilator of extension of scalars vs. the extension the annihilatar

Let $A,B$ be commutative rings with 1, $f:A\to B$ a morphism of rings, $M$ an $A$-module, and $M_B=B\otimes_AM$ the extension of scalars. Then is it the case that $\text{Ann}(M)^e=\text{Ann}(M_B)$? ...
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61 views

Calculate the support of module

Let $A=k[x,y]$ where $k$ is an algebraically closed field and let $M=A/(xy)$ be an $A$-module. I am supposed to calculate $\text{Supp}(M)= \{ P \in \text{Spec}(A) : M_p \not= 0 \}$ where $M_p = S^{-1}...
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223 views

Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$.

Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero. Find the field of fractions of $R$. Find the integral closure of $R$ in it's ...
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1answer
190 views

Splitting of an exact sequence

Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that $x \in \mathfrak m \setminus \mathfrak m^2$. Is it true that $$ \frac{\mathfrak m}{x\mathfrak m} \cong \frac{\mathfrak m}{(x)} \oplus \...
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1answer
106 views

Proving an Equivalent Definition of Shellability

Bruns & Herzog (Cohen-Macaulay Rings) give the following definition of a pure shellable simplicial complex: I am stuck in their proof that condition $(b) \implies (c)$: In the argument ...
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1answer
99 views

$R$ noetherian, $I$ injective $R$-module $\Rightarrow$ $S^{-1}I$ is injective over $S^{-1}R$

I am trying to prove that if $R$ is a noetherian ring, $S$ a multiplicative part and $I$ an injective $R$-module, then $S^{-1}I$ is an injective $S^{-1}R$-module. So far I thought: I reduce to check ...
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1answer
75 views

For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
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107 views

Why would a field *not* be considered a discrete valuation ring?

There are two theorems in Matsumura (p. 78-9) Theorem 11.1 Let $R$ be a valuation ring. Then the following conditions are equivalent: (1) $R$ is a DVR (2) $R$ is a PID (3) $R$ is ...
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167 views

Prove that some canonical homomorphism is injective.

Let $A \not= \{0 \}$ be a Noetherian commutative ring and let $M$ be an $A$-module. Prove that the canonical homomorphism $$M \to \bigoplus_{P \in \text{Ass}(M)} M_p$$ is injective. My question is, ...
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85 views

Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...
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Constant projective dimension of $R/I^i$ for all $i$.

Let $R$ be a local Noetherian ring and $I$ an $R$-ideal. What can we say about the ideal $I$ if the projective dimension of $R/I^i$ for $i \ge 1$ is a finite number which is independent of $i$, i.e., $...
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213 views

Most general version of Hensel's Lemma

Roughly speaking, Hensel's Lemma states that a polynomial $f \in O[X]$ over a certain local ring $(O,\mathfrak{m})$ which factors over the residue field $O/\mathfrak{m}$ into coprime polynomials also ...
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56 views

Spectrum of $\mathbb R[X,Y]$ [duplicate]

Let $A=\mathbb R[X,Y]$. Is it easy to classify the $\operatorname{Spec}A$? I guess it contains at least $(0)$ and $(p)$ for primes $p\in A$ but maybe some else sets. Is it easy to classify those? ...
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1answer
149 views

Zero dimensional Gorenstein ring

Let $(R,\mathfrak m)$ be a zero dimensional Gorenstein ring and $\mathfrak q$ be an $\mathfrak m$-primary ideal of $R$. Then TFAE: 1) $\mathfrak q$ is irreducible, 2) $(0:\mathfrak q)$ is principal, ...
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1answer
107 views

uniqueness of morphism $Spec(K) \to X$ of schemes

let $K$ be a field and $X$ a scheme. I'd like to understand the bijection $Hom_{Sch}(Spec(K), X) \cong \{x \in X | \exists \kappa(x) \to K \}$ That map is given by sending a morphism $f: Spec(K) \to ...
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3answers
138 views

Localisation commutes with taking quotients.

If $A$ is a ring, $S$ a multiplicative set and $I$ an ideal, write $T$ for the image of $S$ in $A / I$. Then $T^{-1}(A/I) \cong S^{-1}A/S^{-1}I$ and in particular, for a prime ideal $P$ we have that $...
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2answers
73 views

Liu, exercise 2.1.4: Minimal prime ideals and nilpotents

In the book "Algebraic Geometry and Arithmetic Curves" Liu wrote in errata that there is a mistake in this problem: Let $A$ be a commutative ring with unit. (a) Let $\mathfrak p$ be a minimal prime ...