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

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0
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1answer
70 views

Prime ideals of infinite depth in Noetherian rings

I'm struggling with the definition of depth of prime ideals given in Atiyah's book: The depth of a prime ideal $p$ is longest strictly increasing chain of prime ideals starting at $p$. Clearly ...
0
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1answer
45 views

An exercise about field automorphisms and ideals.

Consider a field $K$ and the $K$-algebra $K[x_1,\ldots,x_n]$ of polynomials in $n$ variables; $\mathfrak a$ is an ideal of $K[x_1,\ldots,x_n]$ and suppose that there exists a field $L\subseteq K$ ...
3
votes
5answers
135 views

Why does $p(a)=0$ imply $(x-a) \mid p$?

There's something I've never understood about polynomials. Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously, $$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$ The converse of ...
-1
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0answers
97 views

Associated primes of a module

Let $A=k[x,y]$ and $M=A/ x\oplus A/ y$. Define $\operatorname{Ass}M=\{P\in \operatorname{Spec} A\mid M\text{ contains a submodule isomorphic to } A/P\}$, in other words the associated primes or ...
0
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0answers
25 views

Without loss of generality $P_1, . . . , P_s$ contain all elements of $gr_I (R)$ of positive degree, and $P_{s+1}, . . . , P_r$ do not

In the $\underline {Proposition\ 8.5.7}$ of book: Integral Closure of Ideals, Rings, and Modules, (Irena Swanson and Craig Huneke), they say: "Without loss of generality $P_1, . . . , P_s$ contain all ...
1
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2answers
58 views

Atiyah-Macdonald p.108

I don't understand the following lines on p.108 (chapter 10) in Atiyah-Macdonald: Since we have a natural homomorphism $f:A\to \hat{A}$ we can regard $\hat{A}$ as an $A$-algebra and so for any ...
0
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1answer
33 views

F structure of an algebraic set (why is this ring hom injective?)

I'm trying to understand the notion of the field of definition of an algebraic set. Specifically, I'm stuck on page 6 of the book Linear Algebraic Groups by TA Springer. Suppose $F \subset K$ is a ...
0
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0answers
108 views

Find a disassembly for a module.

Let $A=k[x,y,z]$ and $I=(x^2,xy,xz,yz)$. My previous question was how to calculate a primary decomposition of $I$. However there was a part b) added to this exercise, namely to calculate a disassembly ...
2
votes
2answers
45 views

Tensor product of two finitely generated modules

How can I show that if $M$ and $N$ are finitely generated $A$-modules, then so is $M\otimes_AN$? I understand that I have assumption that there are integers $n,m$ such that there are surjections ...
1
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1answer
50 views

$I = (x^2, y^2) ⊂ K[x, y]$; $gin\ (I)=?$

an easy Google search give a lot of results about the definition of generic initial ideal. But all definitions I see, are like this one: I can't use this definition to compute gin(I) even in simple ...
0
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1answer
63 views

Localization of a ring that is not an integral domain

Let $A$ be a commutative ring with unity that is not an integral domain and $\mathcal{P}$ be any prime ideal of $A$. Then I know that $A_{\mathcal{P}}$ is not an integral domain using the ...
0
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0answers
29 views

$I$ is a $J$-primary ideal of $R$ iff $I/L$ is a $J/L$-primary ideal of $R/L$

Let $R$ be a commutative unitary ring and $I$, $J$, $L$ be ideals of $R$ with $L$ proper, $L \subseteq I$ and $L \subseteq J$. A homework question asks to prove that if $R$ is noetherian then $I$ is ...
1
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1answer
36 views

Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra?

(I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following ...
1
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1answer
36 views

The rank of the integral closure as a free module

Let $ O$ be a PID, and let $L$ be a finite separable extension of its quotient field $K$ with degree $n$. Prove that the integral closure of $O$ in the field $L$ is a free module of rank $n$. ...
0
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1answer
37 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
0
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0answers
19 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
1
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1answer
21 views

What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...
0
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0answers
25 views

betti-numbers of Gin(I), generic initial ideal of $I$

here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b): can you help please?
0
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1answer
30 views

Completion of an ideal

On page 109 of Atiyah-Macdonald, the authors let $A$ be a Noetherian ring with an ideal $\mathfrak{a}$. We use the notation $\hat A$ for the $\mathfrak{a}$-dic completion of $A$. They say that we ...
2
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1answer
42 views

For which $n \in \mathbb{N}$ is it the case that every element of $\mathbb{Z}/n\mathbb{Z}$ is strongly associate to an idempotent?

Definition. Call two elements of a commutative ring associates iff each divides the other. Call them strong associates if there exists a unit that can be multiplied by the first to yield the second. ...
4
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1answer
38 views

$L$-Zariski closure of subgroup $SL_n(F)$ as subset of $M_n(F)$ also a subgroup of $SL_n(F)$

Let $F$ be a field, and $SL_n(F)$ be the group of $n \times n$ matrices with determinant $1$. Let $\Gamma \subset SL_n(F)$ be a subgroup. We can consider $\Gamma$ to be a subset of $M_n(F) \cong ...
3
votes
1answer
35 views

Finite generation of modules

Let $M$ be an $R$-Module. Suppose we know that $M$ is finitely generated. Let $X\subseteq M$ be any generating set. Is there a finite subset of $X$ that generates $M$? I stumbled about this when ...
1
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0answers
32 views

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 ...
3
votes
1answer
64 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 ...
2
votes
2answers
111 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 ...
1
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0answers
37 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 ...
-1
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0answers
36 views

Ring of integer valued polynomials is not Noetherian [duplicate]

Let $A := \text{Int}(\mathbb{Z}) :=\{ f \in \mathbb{Q}[x]: f(\mathbb {Z}) \subset \mathbb{Z} \}$. Why $A$ is a non-Noetherian ring ?
4
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1answer
45 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 ...
0
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1answer
41 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 ...
3
votes
1answer
40 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 $R/\operatorname{ann}(M)$. ...
2
votes
1answer
39 views

Matsumura 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 Ass$_A(A/xA)=$Ass$_A(A/x^nA)$ for each $n=1,2,\ldots.$ My question ...
0
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1answer
53 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 ...
3
votes
1answer
88 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 $X$. ...
1
vote
1answer
65 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 ...
5
votes
4answers
128 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 ...
2
votes
1answer
65 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 ...
0
votes
2answers
62 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 ...
6
votes
0answers
51 views

$E \to S$ 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 ...
2
votes
1answer
46 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 ...
1
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2answers
49 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 ...
3
votes
2answers
126 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!)
1
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1answer
32 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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2answers
73 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} ...
0
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0answers
54 views

Picard group of affine scheme of a UFD

In which book/notes can I find proofs of the following facts? 1) Pic(Spec$A)$ is $0$ where $A$ is a UFD. 'Pic' is the Picard group. 2) The invertible sheaves on projective space P$^n(k)$ for $k$ a ...
9
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2answers
338 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 ...
4
votes
1answer
69 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 ...
1
vote
1answer
34 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) ...
1
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1answer
32 views

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

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 ...
0
votes
2answers
73 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 ...
0
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1answer
81 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 ...