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

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4
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1answer
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Monomials not in an ideal

Let $R=\mathbb{R}[x,y]$ denote the commutative ring of polynomials in two variables $x,y$ with real coefficients. Show that for each $k \in \mathbb{N}$ there exists a monomial of degree $k$ not ...
5
votes
0answers
549 views

Integral homomorphism induces a closed map on spectra

I'm trying to prove the following: Let $f:A\rightarrow B$ be an integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
1
vote
1answer
74 views

Isomorphism of polynomial rings implying isomorphism of the coefficient rings [duplicate]

Let $R$ and $S$ be commutative rings. Let $x, y$ be indeterminates, and assume that one has an isomorphism $R[x] \rightarrow S[y]$ (not necessarily mapping $x$ to $y$ of course). Does this imply $R ...
7
votes
2answers
455 views

Coordinate ring in projective space. What are they?

When $X$ is an algebraic variety of affine $n$-space, then the coordinate ring of $X$ are polynomials restricted to $X$. But when $X$ is a variety of projective $n$ space, what are the elements ...
1
vote
1answer
88 views

Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
2
votes
3answers
613 views

Spectrum of polynomial ring

In M. Reid's Undergraduate Commutative Algebra, the author states that if $k$ is an algebraically closed field then $\operatorname{Spec}{k[x]} = \{0\} \cup k$ (page 21). Is this correct? Instead, ...
6
votes
1answer
591 views

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
2
votes
1answer
228 views

What does “Hauptidealsatz” mean in “Krull's Hauptidealsatz”?

What does "Hauptidealsatz" mean in "Krull's Hauptidealsatz"? Thank you very much.
2
votes
1answer
435 views

dimension of an ideal (definition)

Let $A$ be a commutative ring and $I$ an ideal. When we refer to the "dimension" of $I$, what exactly do we mean? Is it the Krull dimension of $A/I$? In particular, i am trying to understand the ...
1
vote
1answer
101 views

Annihilators of Modules

I'm stuck trying to prove that for two $R$-modules $M,N$ ($R$ commutative with a 1), then $$Ann(M+N)=Ann(M) \cap Ann (N)$$ I was trying to do double inclusion, and I can prove the RHS is contained in ...
5
votes
2answers
539 views

Integral closure $\tilde{A}$ is flat over $A$, then $A$ is integrally closed

Question. Let $A$ be an integral domain and $\tilde{A}$ be its integral closure in the field of fractions $K$. Assume that $\tilde{A}$ is a finitely generated $A$-module. I want to prove that if ...
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votes
2answers
197 views

Homogeneous ideals are contained in homogeneous prime ideals

Let $I$ be a homogeneous ideal of a graded ring $S$, $I\ne S$. I want to show that there exists a homogeneous prime ideal which contains $I$. I proved the following: Let $T$ be the set of all ...
2
votes
1answer
81 views

Is localization of a prime ideal still a prime ideal?

Im still new to the topic so this question might seem trivial. But I hope if someone can help explaining to me if a prime ideal $P$ of a domain $A$ is still a prime ideal $P_s$ in the localization ...
4
votes
1answer
111 views

Primary decomposition of power of a prime.

Let $R$ be a commutative Noetherian ring with unit. Suppose $P$ is a prime ideal that is not maximal. How can we go about finding a normal (reduced) primary decomposition of the power of $P$, say a ...
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0answers
133 views

Ideal membership problem for monomial ideals

Hi guys. I'd really appreciate help on understanding the proof for this Lemma above. I'm not sure how we got: "we see that every term on the right side of the equation is divisible by some x^{a(i)}. ...
13
votes
2answers
656 views

What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?

Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find a primary decomposition of $I$. I tried to draw the graph of the variety of $I$ and get a decomposition of ...
2
votes
1answer
251 views

degree lexicographic monomial ordering

With respect to deglex X>Y, what would the leading monomials of these polynomials be? $f_1=XY^3-X^2$ and $f_2=-X^3Y^3-4X^2Y^3+3X^2Y$ My understanding is that you prioritise X over Y based on their ...
28
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0answers
478 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
1
vote
1answer
158 views

Not primary ideal having a prime radical

Find an example of a ring $A$ and an ideal $I$ such that I is not primary but if $fg\in I$, then $\exists n\in \mathbb{N} $ such that $f^{n}\in I $ or $g^{n}\in I$.
1
vote
1answer
189 views

Irreducible polynomials and affine variety

Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite.
7
votes
1answer
143 views

When does “second annihilator” of a (principal) ideal equal the ideal itself

Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $\operatorname{Ann}_R(\operatorname{Ann}_R(r))=Rr$ or $\sqrt{\operatorname{Ann}_R(\operatorname{Ann}_R(r))}=\sqrt{Rr}$ hold? I ...
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0answers
68 views

A counterexample for $\operatorname{Ass}(M_1+M_2)$ [duplicate]

$\newcommand{\Ass}{\operatorname{Ass}}$ Let $A$ be a Noetherian ring and let $M$ be an $A$-module. Suppose $M=M_{1}+M_{2}$, then we have $\Ass(M)\supset \Ass(M_{1})\cup \Ass(M_{2})$. What is an ...
3
votes
1answer
90 views

square system of polynomial equations having infinite number of solutions

Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
3
votes
1answer
94 views

Small question about a proof of Hilbert's Basis Theorem

I am currently going going through the proof of Hilbert's Basis Theorem: http://www.maths.usyd.edu.au/u/de/AGR/CommutativeAlgebra/pp806-850.pdf (it starts on slide 832) On slide 836-837 he makes the ...
5
votes
1answer
1k views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
3
votes
2answers
131 views

$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact, $M''$ flat. Why is $M$ flat $\Leftrightarrow M'$ flat?

Let $A$ be a commutative ring with identity, and let \begin{align} 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 \end{align} be an exact sequence of $A$-modules with $M''$ flat. ...
3
votes
1answer
123 views

All the Associated Primes are minimal.

Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question. 1) Suppose $\operatorname{Ass}(R/I)$ are all ...
3
votes
1answer
455 views

Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and non-zero-divisors (exercise in SICA)

In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written: (1) Consider the two rings $$ A=\mathbb C[x,y]/\langle x^2-y^3\rangle\text{ and } B=\mathbb ...
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0answers
67 views

Noetherian localizations and extra-condition implies Noetherian

I'm trying to solve this question but I'm stucked: If a ring $R$ satisfies the following two conditions: i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is ...
12
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1answer
152 views

How does Local Cohomology detect UFD?

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a ...
1
vote
1answer
299 views

What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
8
votes
1answer
257 views

Gröbner basis and generating set

I have come across the following past exam question... Define an ideal $J:=(z^2x+y^2-2y,x^3+y^3+z^3,x^2+2z^2) \subseteq \mathbb{Q}[x,y,z]. $ Compute a generating set for $J \cap \mathbb{Q}[y]$. ...
9
votes
2answers
184 views

What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
3
votes
3answers
768 views

Zero-dimensional ideals in polynomial rings

I have the following past exam paper question, a similar sort of question seems to come up every year. And I'm completely lost with it... Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated ...
2
votes
1answer
106 views

Show that $M=\bigcap_{\mathfrak{p}\in\operatorname{Spec}(R)}M_\mathfrak{p}=\bigcap_{\mathfrak{m}\in\text{Max}(R)}M_\mathfrak{m}$ for certain $M$.

$\newcommand{\Spec}{\operatorname{Spec}}$ $\newcommand{\mSpec}{\operatorname{Max}}$ This is a homework from my algebra course. I am in a situation where I think I have found a solution, though ...
3
votes
1answer
58 views

Difficulty Understanding Primary Modules

I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
6
votes
2answers
353 views

Intuition behind Direct limits

Let $R$ be a commutative ring and $x\in R$ be a nonzero divisor. Then i know that the direct limit of $R\mapsto R\mapsto R\mapsto\cdots $, where each map is multiplication by $x$ is $R_x$, the ...
3
votes
1answer
152 views

Flatness versus vanishing of Tor-groups for a non-finitely generated module

This is something I should probably know, but it is escaping me at the moment. Let $A$ be a commutative noetherian ring. The following corollary of Nakayama's lemma is well-known (for instance, this ...
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2answers
188 views

For a nilpotent $r\in R$, show $r$ is in every prime ideal and that $1-sr$ is a unit for all $s\in R$.

Let $p \subset R$ be a prime ideal. Prove that for any nilpotent $r \in R$, it follows that $r \in p$. One of my classmates told me to use induction. Also, show that for all $s \in R$, $1-sr$ ...
3
votes
1answer
82 views

Injective dimension is locally finite but not globally

Let $A$ be a commutative ring. Could someone provide me an example where $\operatorname{id}_{A_{\mathfrak p}}(M_{\mathfrak p})$ is finite for all $\mathfrak p\in \operatorname{Spec}(A)$, but ...
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vote
0answers
64 views

robust computation of Groebner basis

I am trying to solve numerically polynomial systems of equations, over the reals. I am coming across the following phenomenon: let's say that i have a system of 7 equations with 7 unknowns. I am using ...
5
votes
1answer
237 views

Vakil 14.2.E: $L\approx O_X(div(s))$ for s a rational section.

I am working through Vakil's Ch 14 (march2313 version) on invertible sheaves and am having trouble on 14.2.E. The question (in notation to be defined) is this: how do I show that each point in the ...
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0answers
48 views

constructing a sum of squares modulo an ideal

This question refers to the proof of Theorem 7.3, p. 98 of the pdf http://math.berkeley.edu/~bernd/cbms.pdf. The statement of the theorem and its proof do not depend on what precedes them. Let $I$ be ...
4
votes
1answer
188 views

About Artinian Rings

I'm studing commutative algebra by the text of Atiyah and Macdonald, and a doubt come at me and I can not prove neither find a counterexample, the problem is: If a ring (commutative with identity) ...
5
votes
2answers
182 views

Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$

I'm studying for my algebra quals this fall and keep encountering problems like the following: List all the ideals of $\mathbb{Z}[x]/(16, x^3)$. or List all the primes of ...
1
vote
1answer
53 views

Finding a presentation of $A$-algebra $B$

Find a presentation of the $A$-algebra $B$, where $B=\mathbb{Z}[1/2]\subseteq \mathbb{Q}$ and $A= \mathbb{Z}$. I want to prove it but I can't understand what want to me! Please describe to me.
6
votes
1answer
76 views

Function field question from Silverman's AEC

Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring. Then he states Proposition 1.7 (the intrinsic characterization of ...
7
votes
1answer
465 views

Maximal ideals in the algebra of continuously differentiable functions on [0,1]

This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
3
votes
1answer
106 views

Computing a rational function at a point in terms of a uniformising parameter

I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
13
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0answers
729 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...