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

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1answer
82 views

Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the ideal $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime ideal? If so, what is its height? I'm stuck trying to show that $f$ is ...
4
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1answer
240 views

How to find a finite set of generators for $I \subset k[x_1, …, x_n]$

Suppose that we have a set of polynomials $f_1, ..., f_m \in C = k[x_1, ..., x_n]$ where $k$ is an algebraically closed field, and let $V$ denote the set on which they all simultaneously vanish. ...
7
votes
1answer
428 views

Is the image of a tensor product equal to the tensor product of the images?

Let $S$ be a commutative ring with unity, and let $A,B,A',B'$ be $S$-modules. If $\phi:A\rightarrow A'$ and $\psi:B\rightarrow B'$ are $S$-module homomorphisms, is it true that $$\operatorname{im}(\...
6
votes
2answers
165 views

Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space

Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ ...
0
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1answer
95 views

Is inclusion of a prime ideal into a different prime ideal possible?

Let $A$ be a commutative ring with identity. Let $p, q$ be two distinct prime ideals. Is it possible that $p \subseteq q$?
4
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2answers
383 views

Spectrum of finite $k$-algebras

Let $k$ be a field and $A$ be a finite $k$-algebra. How does one quickly see that $Spec(A)$ is a finite set? Further, is it true that the cardinality of $Spec(A)$ is equal to $dim_k(A)$?
5
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1answer
624 views

Generating set for sum of two ideals

Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + ...
5
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1answer
140 views

Is $f$ reduced if and only if the derivations $\gcd(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=1$ under some conditions?

I have encountered the following problem. I have no ideas to prove it or disprove it. Suppose that $f\in \mathbb{C}[x,y]$, $f(0,0)=0$, $\frac{\partial f}{\partial x}(0,0)=0, \frac{\partial f}{\...
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5answers
3k views

Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian.

I am revising for my Rings and Modules exam and am stuck on the following two questions: $1.$ Let $M$ be a noetherian module and $ \ f : M \rightarrow M \ $ a surjective homomorphism. Show that $f ...
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3answers
284 views

Example where $\operatorname{grade}(I,M)>\operatorname{height} I$

Let $I$ be an ideal of a noetherian ring $R$ and let $M$ be a finite $R$-module. We need to show if $I$ is generated by $n$ elements, then $\operatorname{grade}(I,M)\le n$. Could any one give an ...
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1answer
133 views

How to show that the coordinate ring of a finite set of points in projective space is Cohen-Macaulay?

Could anyone give me a hint how to show this one: Let $V$ be a finite set of points in projective space. How to show that the coordinate ring of $V$ is Cohen-Macaulay?
4
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1answer
326 views

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here ...
6
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1answer
280 views

Is the additive semigroup of natural numbers the multiplicative semigroup of a ring?

$\mathbb N$ will denote the set $\{0,1,2,\ldots\}.$ The semigroup $(\mathbb N,+)$ doesn't have a zero element. $\mathbb N^0$ will denote the semigroup $\mathbb N$ with zero adjoined, that is the set $\...
4
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1answer
521 views

Why is this ideal projective but not free?

Let $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$. How can I prove that $I$ is projective but not free?
5
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1answer
135 views

What is the injective hull of $\mathbb{C}(x,y)/\mathbb{C}[x,y]$ ?

What is the injective hull of $\mathbb{C}(x,y)/\mathbb{C}[x,y]$ as a $\mathbb{C}[x,y]$-module? Is it isomorphic to any familiar module?
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1answer
117 views

Doubt about proof of completeness

Let me refer you to: http://www.math.iitb.ac.in/atm/caag1/balwant.pdf Part (2) line $4$ can someone please explain why $x_{ni}=x_{n(m)i}$ for all $i \leq m$ and $n \geq n(m)$. I know it should ...
5
votes
2answers
88 views

Integrally closed with roots of identity

Let $\lambda_1,...,\lambda_n$ be roots of unity with $n\geq 2$. Assume that $$\frac{1}{n}\sum_{1}^{n}\lambda_i$$ is integral over $\mathbb{Z}$. Show either $\sum_{1}^{n}\lambda_i=0$ or $\lambda_1=\...
8
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2answers
552 views

Approximation Lemma in Serre's Local Fields

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated. Approximation Lemma Let $k$ be a positive integer. For every $i$, $1\...
3
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3answers
106 views

$N$ submodule of $M$ and $N \cong M$ does not necessarily imply that $M=N$

Let $M, N$ be $A$-modules with $A$ being a commutative ring. Suppose that $N$ is a submodule of $M$ and also that $N$ is isomorphic to $M$. According to my understanding this does not necessarily ...
3
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1answer
83 views

$S^{-1}M \cong S^{-1}N$ does not imply $M \cong N$

Let $M, N$ be $A$-modules, where $A$ is a commutative ring with identity. Let $S$ be a multiplicative subset of $A$ that contains no zero divisors and contains the identity of $A$. I am looking for a ...
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1answer
603 views

Symmetric power of vector space

Let $V$ be a vector space over a field $k$ of char. zero and denote by $Sym^n_k V$ its $n$-th symmetric power over $k$. Now I simply want to know what $Hom_k(V,Sym^n_k V)$ is for $n \geq 2$. To be ...
7
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2answers
270 views

Isomorphism between two localizations

I am doing exercise 3.23 in Atiyah Macdonald and in the first part of the problem they ask to show that the ring $A_f = S^{-1}A$ where $S = \{1,f,f^2 \ldots \}$ depends only on the choice of the basic ...
6
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2answers
477 views

Artinian if and only if Noetherian

Let $R$ be a ring (commutative, with identity), $m$ a maximal ideal and $M$ an $R$-module. Suppose $m^nM=0$ for some $n>0$. Then $M$ is Noetherian if and only if $M$ is Artinian Do you have any ...
4
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2answers
162 views

An equivalent condition for an element to be integral

Let $R$ be a noetherian domain, $Q$ its field of fraction and $u\in Q$. Could you help me to prove that $u$ is integral over $R$ if and only if there exists $r\in R$ $r\neq0$ and $ru^n\in R$ for ...
2
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1answer
297 views

How much connection is there between Commutative Algebra and Algebraic Topology?

How much connection is there between Commutative Algebra and Algebraic Topology? I am looking for general highlights, not complex details.
2
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0answers
92 views

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Particularly I'd like to know the formulation thereof which concerns the kernel of a surjective ring ...
2
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1answer
120 views

Why is this homomorphism an isomorphism?

Let $R$ be a commutative ring with identity. Suppose $R=(r_1,\ldots,r_k)$. Take an homomorphism of $R$-modules: $f:M\rightarrow N$. Suppose that the function $\frac{f}{1}:M_{r_i}\rightarrow N_{r_i}$ ...
6
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2answers
1k views

Noetherian and Artinian modules

I don't understand clearly what is meant by Noetherian and Artinian modules. I tried explaining it to myself using the definitions but it is still not clear. Can someone please help explain it to me ...
6
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0answers
307 views

Showing an ideal is prime in polynomial ring

Let $k=\mathbb{C}$ and let $J$ the ideal $(xw-yz,y^{3}-x^{2}z,z^{3}-yw^{2},y^{2}w-xz^{2})$. I want to see why $J$ is a prime ideal in $k[x,y,z,w]$. I know that $Z(J)$ (the zero set of $J$) is ...
3
votes
5answers
295 views

units in commutative rings

An element $a$ of a ring $R$ is called a unit if it has a two sided inverse under multiplication; that is, if there exists $b \in R$ such with $ab = ba = 1_R$. How would you show that if $R$ is ...
6
votes
3answers
681 views

$R^n \cong R^m$ iff $n=m$

How can i show that two $R$-modules of finite rank are isomorphic if and only if they have the same rank, i.e., $R^n \cong R^m$ iff $n=m$.
11
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2answers
502 views

Showing a UFD which is not a PID must have a nonprincipal maximal ideal.

Given that $R$ is a UFD which is not a PID, I want to show that $R$ must have a nonprincipal maximal ideal. I tried several methods, including Zorn's lemma but didn't get anywhere. Any suggestions ...
4
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2answers
147 views

What do ideals of a ring say about its inner structure?

Could people with knowledge in Commutative Rings elaborate on this sentence from the Wikipedia article (Ideals and Factor Rings subsection, first sentence): The inner structure of a commutative ...
2
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0answers
297 views

Inductive proof of a version of Nakayama's lemma

In Matsumura's 'Commutative Ring Theory', the following version of NAK is shown: If $M$ is a finitely generated $A$-module, $I\subseteq A$ an ideal s.t. $IM=M$, then there exists an $a\in A$ with $...
4
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1answer
169 views

If an ideal $I$ contains a non-zero-divisor then $\mathrm{End}_R(I)$ is commutative

How can I prove that if $I$ is an ideal of a commutative ring $R$ that contains a non-zero-divisor then $\mathrm{End}_R(I)$ is commutative?
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0answers
43 views

$S^{-1}\left(A^{M \times N} \right) \subsetneq \left(S^{-1}A\right)^{(S^{-1}M) \times (S^{-1}N)}$

Let $M, N$ be $A$-modules, where $A$ is a commutative ring. Let $S$ be a multiplicative subset of $A$. I want to study the relation between $S^{-1}\left(A^{M \times N} \right)$ and $\left(S^{-1}A\...
38
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2answers
2k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
3
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2answers
350 views

Quotient of an affine variety by a finite group coincides with topological quotient as a point set?

I have just read the construction of the quotient of a closed subset $X$ of affine space by a finite group $G$ of automorphisms of $X$, in Shafarevich, Basic Algebraic Geometry I. Shafarevich gives ...
3
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1answer
253 views

Ring of Fractions

Let $A$ be an integral domain. If $S=A-\left\{0\right\}$, then $S^{-1}A$ is the field of fractions of $A$. What is the problem if we actually take $S=A$? From what i see, in that case $0/0=1/1$ and ...
2
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1answer
427 views

Tensor product of a finitely generated modul and a finite length module is finite length

Let $R$ be a commutative ring and $M,N$ $R$-modules finitely generated with $M$ of finite length. How can I prove that $M\otimes_R N$ is of finite length?
4
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1answer
287 views

Turning the tensor product of algebras into an algebra

Let $B, C$ be $A$-algebras, where $A$ is a commutative ring, i.e. $B, C$ are rings and we have ring homomorphisms $f:A\rightarrow B, g:A \rightarrow C$. Since both $B, C$ are $A$-modules, we define $D=...
4
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2answers
399 views

Showing polynomials in $k[x_1, \ldots , x_n]$ are irreducible

It is often the case when I wish to show a particular polynomial in $k[x_1, \ldots ,x_n]$ is irreducible. Assuming that the polynomial is sufficiently friendly (i.e. one I would encounter as part of a ...
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1answer
86 views

Understanding a morphism of modules by properties of the induced residue field homomorphism

Let $A$ be a reduced local Noetherian ring, and $\phi: M\to N$ a morphism of finitely generated free $A$-modules. For all $\mathfrak{p}\in\text{Spec}(A)$, let $k(\mathfrak{p})=A_\mathfrak{p}/\mathfrak{...
1
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1answer
272 views

indecomposable module which is not cyclic

In Etingof's notes entitled "Introduction to Representation Theory," he asks the reader to produce an example of an indecomposable module which is not cyclic (Problem 1.25(c)). The exercise even comes ...
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0answers
513 views

An algebra of finite type over a field is a Jacobson ring.

Let $R$ be an algebra of finite type over a field $k$. I want to show that $R$ is Jacobson, i.e. that any prime ideal $\mathfrak{p}$ in $R$ is an intersection of maximal ideals. I am not getting very ...
14
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3answers
2k views

About the localization of a UFD

I was wondering, is the localization of a UFD also a UFD? How would one go about proving this? It seems like it would be kind of messy to prove if it is true. If it is not true, what about ...
1
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2answers
202 views

$(B \otimes C) \otimes (D \otimes E)$ is isomorphic to $B \otimes C \otimes D \otimes E$

Let $B, C, D, E$ be $A$-modules. Is there a way to show that $(B \otimes C) \otimes (D \otimes E)$ is isomorphic to $B \otimes C \otimes D \otimes E$ using the result that $(M \otimes N) \otimes P$ ...
3
votes
1answer
248 views

Not a radical ideal

Is there a method to compute the radical of an ideal? for example take $J=(xw-y^{2},xw^{2}-z^{3}) \subset k[x,y,z,w]$. I want to show $J$ is not radical, I guess the idea is to add and substract terms ...
7
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0answers
661 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
5
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1answer
196 views

Tensor-free proof that for finite modules over reduced Noetherian rings, locally free = projective

Is there an elegant tensor-free proof of the fact that over a reduced Noetherian ring $A$, every finitely-generated $A$-module which is locally free, is projective? EDIT: I would be content with the ...