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

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3answers
80 views

Ideal Generated by Three Elements in Polynomial Ring [duplicate]

How would one prove that the ideal $(xy,xz,yz)$ of $k[x,y,z]$ for some field $k$, cannot be generated by two polynomials. In other words, prove: $$(xy,xz,yz) \neq (f,g)\; \forall f,g \in ...
3
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1answer
56 views

Rank of projective module defined as the smallest $n$ such that $P$ is a direct summand of $R^n$

Over a commutative ring $R$, the rank of a projective module $P$ is defined by looking at the map $\text{rank}(P) : \text{Spec}(R) \rightarrow \mathbb{N}_0$ given by $\mathfrak{p}\mapsto ...
0
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1answer
24 views

Decomposition of a polynomial over generators of an ideal

Let $f$ be a polynomial in six variables, say, over complex numbers, and $l_1$, $l_2$ are some linear forms in the same variables. If I know that polynomial $f$ belong to the ideal generated by $l_1$ ...
0
votes
0answers
21 views

Primality of homogeneous ideal

Let $R$ be the polynomial ring over the finite field $\mathbb{F}_p$ with $n$ variables. Let $I$ be an ideal of $R$ generated by homogeneous polynomials whose coefficients are 1 or -1. Are there any ...
0
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1answer
52 views

Primary decomposition of $(XY,(X-Y)Z)$ in $k[X,Y,Z]$

How to find the primary decomposition of $I=(XY,(X-Y)Z)$ in $R=k[X,Y,Z]$? It has minimal primes $(x,y),(y,z),(z,x)$. I tried to calculate $J=S^{-1}I\cap R$, where $S=R-(x,y)$, but it seems ...
0
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1answer
45 views

Is a graded module over a graded ring zero when all of it's graded localizations at graded primes not containing the irrelevant ideal are zero?

Let $M$ be a graded module over an $\mathbb{N}$-graded ring $S$ and $S_+$ be the ideal of positive degree elements. Is it true that $M=0$ iff the homogeneous localization $M_{(\mathfrak p)}=0$ for ...
0
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1answer
66 views

Hartshorne Chapter II exercise 5.7 on Invertible sheaves

I'm working on part c) which is to prove that for a Noetherian scheme $X$, a coherent sheaf $\mathscr{F}$ is invertible (locally free of rank 1) iff there exists a coherent sheaf $\mathscr{G}$ such ...
1
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1answer
51 views

Factorization in noetherian domains

I changed the title (and the body) of this question page, since user26857 provided a nice answer for my original question in a more general setting. Here's what the accepted answer below provides: ...
0
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1answer
44 views

Lifting a direct summand of a free module

Suppose $R$ is a commutative ring, $I\subseteq R$ a principal ideal, and we're given split short exact sequences $ R \to R^n \to R^{n-1}$ and $ R/I \to (R/I)^n \to (R/I)^{n-1}$ the first inducing ...
0
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0answers
25 views

Injective map which on quotients is the inclusion of a direct summand

suppose we are given a commutative ring $R$, a principal ideal $I$ and an injective ring map $R \stackrel{f}{\longrightarrow} R^n$, which on quotients $R/I \longrightarrow (R/I)^n$ is the inclusion of ...
0
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0answers
73 views

Irreducible ideals that are not primary.

In my advanced algebra course I've heard that in a noetherian (commutative) ring every irreducible ideal is primary. Can you give a counter example in a non noetherian ring? I've been lookin' ...
1
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1answer
48 views

Showing regularity by the Auslander-Buchsbaum formula

Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ with $\operatorname{gl.dim}(R) < \infty$. According to this Wikipedia article it follows from the ...
1
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1answer
61 views

Koszul Homology vs Koszul Cohomology

Let $R$ be a ring and $x \in R$. The Koszul complex $K_\bullet(x)$ is then $0 \rightarrow R \stackrel{x}{\rightarrow} R \rightarrow 0$. Given $x_1,\dots,x_n \in R$ the Koszul complex ...
0
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1answer
17 views

Dimension of quotients of a discrete valuation domain

I'm learning some properties of discrete valuation rings (DVR's further for geometrical use). By the way, a domain $R$ is said to be a DVR if there exists the so called uniformizing parameter $t$ such ...
6
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1answer
80 views

What are the points of some schemes?

Let $X=\operatorname{Spec}\mathbb{C}[x,y,t]/(xy-t)$, $Y=\operatorname{Spec}K[x,y]/(xy-t)\rightarrow \operatorname{Spec}K$ and $Z=\operatorname{Spec}R[x,y]/(xy-t)\rightarrow \operatorname{Spec}R$, ...
8
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2answers
103 views

${\rm Hom}_R(M, R/M) =\{0\} \implies R$ is a field.

Let $R$ be a local ring with maximal ideal $M$. Suppose $M$ is finitely generated. Prove that if ${\rm Hom}_R(M, R/M) =\{0\}$, then $R$ is a field. ${\rm Hom}_R(M, R/M)$ stand for the group of ...
0
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1answer
40 views

Closedness and going up property

Let $f: A\rightarrow B$ be a homomorphism of commutative unital rings. The problem is to show that if $f$ has going-up property and $\text{Spec }B$ is Noetherian topological space then $f^*: ...
0
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3answers
76 views

If $M$ is a flat $R$-module, is $M/IM$ a flat $R/I$-module?

Let $R$ be a Noetherian (local) ring, and let $M$ be a finitely generated, flat $R$-module. Further, let $I$ be an ideal of $R$. Question: Is $M/IM$ flat over $R/I$?
0
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1answer
47 views

A question on graded rings

For a ring $A$ and an ideal $\mathfrak{a}$ of $A$, Atiyah-Macdonald define $$A^*=\bigoplus_{n=0}^\infty \mathfrak{a}^n$$ and claim that it is a graded ring on p. 107 of their commutative algebra book. ...
0
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0answers
71 views

A question involving the module of differentials

Let $B$ be a local ring. Let $k$ be its residue field. Do we need $B$ to contain a copy of $k$ in order for the following to be true: $$\operatorname{Hom}_{k}({\Omega_{B/k}\otimes_{B} ...
0
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1answer
25 views

every ideal that is not intersect $S$ is prime ideal?

every ideal that is not intersect $S$ where $S$ is multiplicative closed is prime ideal? I know that maximal ideals among those are prime ideals. But what about other ideals that is not intersect $S$. ...
0
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0answers
52 views

Tensoring an exact sequence of $R$-modules with $R/x$

Let $R$ be a commutative ring with an $R$-module $M$, and let $x \in R$ be an $M$-regular element. Then tensoring any short exact sequence $0 \to B \to A \to M \to 0$ with $R/x$ yields a short exact ...
1
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1answer
45 views

Does $ax\in\mathfrak{m}I$ with $x\in I\setminus\mathfrak{m}I$ and $a \in R$ imply $a\in\mathfrak{m}$ for an invertible fractional $R$-ideal $I$?

Let $R$ be an integral domain, $\mathfrak{m}$ a maximal ideal of $R$, and $I$ an invertible fractional $R$-ideal. If $x \in I \setminus \mathfrak{m}I$ and $a \not\in \mathfrak{m}$, do we have $ax ...
1
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2answers
53 views

It's true that a valuation ring $R$ in the quotient field of a normal ring $A$ contain $A$?

Let $A$ be a finitely generated $k$-algebra ($k$ algebraically closed) of dimension one, integrally closed in its quotient field $K$. Let $R\subseteq K$ be a valuation ring. It's true that $A\subseteq ...
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0answers
36 views

Prove there are some elements in a commutative module [duplicate]

let R be a commutative ring and I is an ideal in R and also M is finite generating module on R. If $\varphi:\:M \to M$ be a homomorphism and $\varphi(M)\subset IM$ .Prove there are some elements ...
2
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0answers
58 views

Non-Noetherian subring of F[X,Y]

I am trying to prove that, for a given field $F$, the subring $$R:=\{p(X,Y)=\sum c_{ij}X^iY^j \in F[X,Y] : c_{0j}=c_{j0}=0 \text{ whenever } j>0\}$$ of $F[X,Y]$ is not Noetherian. I think I ...
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2answers
65 views

How to show rational function field of an affine subvariety with dim>0 is not algebraically closed?

I do not know how to show the following statement. If $X\subset A^n$ is an irreducible subvariety, $\dim X>0$, then the rational function field of $X$, $K(X)$ is not algebraic closed. What ...
6
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1answer
66 views

Generalizing the Big Omega function to Integral Domains

The $\Omega(n)$ function counts the total number of prime factors of $n$ counting multiplicity. Obviously, this definition extends to any Unique Factorization Domain. I have two follow up questions: ...
1
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2answers
79 views

Prime ideals in $k[x,y]/(xy-1)$.

Let $k$ a field. Let $f$ be the ring injective homomorphism $$ f:k[x] \rightarrow k[x,y]/(xy-1)$$ obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ ...
2
votes
1answer
95 views

Elementary motivations for free resolutions

Let $M$ be a finitely generated module over a Noetherian ring $R$ which admits a finite free resolution $0 \to F_n \to \dots \to F_0 \to M \to 0$. There is no doubt that knowing such a resolution is ...
0
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1answer
46 views

On local ring homomorphisms

Suppose I have two local rings $A$ and $B$, and suppose I have $\phi : A \rightarrow B$, which is a ring isomorphism. Does it follow then that $\phi$ is a local ring homomorphism? The point of ...
2
votes
2answers
43 views

Characterization of ideals of algebra of continuous functions on a compact space.

I was reading this planetmath page on the connections between the topology on a compact Hausdorff topological space $X$ and the maximal ideals on the algebra of continuous functions $C(X)$ on $X$, ...
1
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0answers
36 views

Commutative diagram of algebras in Atiyah and Macdonald.

On page 31 of Atiyah and Macdonald, there is a commutative diagram. It essentially says that if $B$ and $C$ are $A$-algebras with ring morphisms $f:A\to B$ and $g\colon A\to C$, and $D=B\otimes_A C$ ...
0
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0answers
35 views

Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16

This question is from the Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16(a). Let $x_1,..., x_n, y, z$ be elements of $R$ such that ideals $(x_1,..., x_n,y)$ and $(x_1,..., x_n,z)$ are ...
2
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1answer
56 views

$I(Y) = \{ p(x,y,z) \in k[x,y,z] \mid p (t,t^2,t^3) = 0, \forall t \in k \}$ is prime

I've been working on the following problem from Hartshorne: Let $Y\subseteq \mathbb{ A }^3 $ be the set $Y = \{(t,t^2 , t^3) \mid t \in k \}$. Show that $Y$ is an affine variety of dimension $1$. To ...
1
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1answer
73 views

Finitely generated prime ideal and annihilator

Suppose $R$ is a commutative ring, $P$ is a prime ideal of $R$, $P$ is finitely generated, and $\operatorname{Ann}(P)=0$. Show that $$\operatorname{Ann}(P/P^2)=P.$$ These are my efforts: ...
1
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1answer
41 views

vanishing of an Ext-Functor for a finite graded module of positive grade over a polynomial ring

Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $M$ be a finitely-generated, graded $S$-module, such that there exists a homogeneous $M$-regular element $\xi ...
2
votes
1answer
51 views

Algebraic Curves: Valuation at a point

I would like to understand the notion of valuation on the local ring of a curve at a point. In the Book The Arithmetic of Elliptic Curves in chapter 2, Example 1.3 $$V:\ Y^{2}=X^{3}+X$$ I don't ...
1
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1answer
46 views

A relation in a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module s.t. $M=IM$. How to prove: $$\exists a \in I \text{ such that } (1-a)M=0. $$ I tried to solve: ...
1
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3answers
57 views

Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors. [duplicate]

Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors. I begin to assume that $A$ has no zero divisors but I don't know continue... \ How would be this proof? thanks! :)
1
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0answers
44 views

Rings and modules

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-modules. Now let $I$ be a proper ideal of $R$. What is the structure of $I$. Is it true that $I$ is a direct sum of cyclic ...
0
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1answer
66 views

Prove that a subset is a finitely generated subring

Consider $\mathbb{A}^2$ with $\rho : (x, y) \mapsto (-x, -y)$. Can anyone help me prove that $S = \{f \in \mathbb{C}[x, y] : f \circ \rho = f\}$ is a finitely generated subring? Also, can $S$ be ...
3
votes
1answer
76 views

Atiyah-Macdonald 5.2

Exercise 5.2 in Atiyah-Macdonald asks to show the following: "Let $A$ be a subring of a ring $B$ such that $B$ is integral over $A$, and let $f: A \to \Omega$ be a homomorphism of $A$ into an ...
1
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1answer
111 views

A relation involving an endomorphism of a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module. (Usually in this problem $R$ includes $1_R$.) Let $\phi : M \to M$ be an $R$-homomorphism, and ...
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votes
2answers
35 views

Prove that T is not a zerodivisor in A[T]

Let A be any ring, consider the polynomial ring A[T]. Prove that T is not a zerodivisor in A[T]. Generalise the argument to prove that a monic polynomial $$ f=T^n+a_{n-1}T^{n-1}+\dots+a_0 $$ is ...
7
votes
1answer
115 views

Tensor Product, Exterior Power and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $$0\to K\to M\otimes M\to M\wedge M\to 0$$ ...
2
votes
2answers
77 views

Going-up and going-down theorems: motivation

I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book. I'm wondering if anyone could give me some basic facts/examples to help me understand why ...
1
vote
1answer
47 views

Primary ideal exercise

I have an exercise about the properties of primary ideal. It's Exercise 15.17 of "Step in commutative algebra", R. Y. Sharp. Let $(A,\mathfrak{m})$ be a local ring and $I$ be a proper ideal of $A$. ...
1
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0answers
40 views

A query on Veronese mapping

The Veronese mapping defined as usual on some $P^n$. Then it is certainly regular. I want to prove that the inverse map to this map is also regular. I have an idea to use projections with ...
1
vote
2answers
24 views

$rad(I)=\cap_{I\subset P,~P~prime}P$

$R$ commutative ring with unity. $I$ R-ideal. Then $rad(I)=\cap_{I\subset P,~P~prime}P$. That is, the radical of $I$ is the intersection of all prime ideals containing $I$. There is a proof of this ...