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

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Show that $A$ is integral over $A^G$

I'm working through some problems in Atiyah - MacDonald, and I need a bit of help understanding the solution of a particular problem. Here is the problem: $G$, a finite group over which acts by ...
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1answer
47 views

localized at associated prime of an ideal [duplicate]

The problem is as follows: Let $I\subseteq J$ be ideals in a Noetherian ring. Show that if $I_{p}=J_{p}$ for every associated prime $p$ of $I$,then $I=J$. It seems reasonable to consider ...
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1answer
34 views

normalization of a curve, simplest example

I am learning about normalization of nodal curves and I am trying to understand the simplest example: $xy=0$ As far as I understand its coordinate ring is $k[x]\oplus k[y]$ (let $k$ be an ...
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0answers
35 views

Non-zero ideal in algebraic integers generated by two elements

I've been doing past questions for my exams next week and would like to check an answer: Let $I$ be a non-zero ideal of the algebraic integers and let $0\neq a \in I$. Show that $\exists b \in I$ ...
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25 views

How do you compute the inverse of $f+I \in S/I$, where $S=K[x_1, \dots, x_n]$ and $I$ an ideal of $S$?

For example, if $f=y+x+1$, the inverse of $f+I$ exists, I want to know what's the easiest and fastest way to find the inverse using a CAS (preferably Singular or CoCoA), if necessary, and using the ...
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41 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. ...
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1answer
47 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
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2answers
33 views

Integrally Closed domain and Principal Ideal

Let $R$ be an integrally closed local domain. Suppose there is a $y\in I^n$ such that $yI^n=I^{2n}$ for some $n$. I would like to prove that $I^n=(y)$. Source: The above question comes from the ...
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1answer
51 views

What does $(0:x)$ mean?

The following excerpt is from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry" on pg. 424 We can decide whether an element $x\in R$ is a nonzerodivisor from the homology of ...
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1answer
86 views

Can I use Krull dimension to test if a sequence of polynomials is regular?

A sequence $(f_1, \ldots, f_n)$ of elements of a commutative ring $R$ is said to be regular if for each $i$, $f_i$ is not a zero divisor in $R/(f_1, \ldots, f_{i-1})$. Call a sequence dimension ...
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2answers
205 views

A criterion of flat modules

Let $R$ be a commutative ring and $M$ an $R$-module such that for every ideal $I \subset R$ the natural map $I \otimes_R M \rightarrow IM$ is an isomorphism. Why is $M$ flat ? This result is ...
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1answer
45 views

Is every “prefield” a field?

Definition 0. Call a poset $P$ well-ranked iff it is well-founded, and for all $x \in P$, we have that any two maximal subchains in the lowerset generated by $x$ have the same length. ...
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1answer
25 views

Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
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1answer
21 views

Regular element of a Noetherian ring [duplicate]

Let $R$ be a Noetherian ring and $x\in R$ an $R-\mathrm{regular}$ element. Show that $\mathrm{Ass}_R(R/(x^n))=\mathrm{Ass}_R(R/(x))$ for every $n\geqslant 1$. Let $M$ be an $R-\mathrm{module}$. An ...
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2answers
120 views

Powers of prime ideals

I was reading through Atiyah-MacDonald and they mention that if a ring $A$ is a Noetherian domain of dimension 1 has the property that every primary ideal is equal to the product of a prime ideal ...
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0answers
23 views

Ideal homogeneous cousin and equivalence

Let $R$ be a graded ring and $I$ ideal in $R$ and homogeneous. $I$ is prime if and only if for all $a, b\in R$ homogeneous such that $ab\in I$ then $a\in I$ or $b\in I$. Let $ab\in I$ and $a ...
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29 views

What came first: pythagoras number or pythagorean fields? [migrated]

Which concept was first introduced: the pythagoras number of a field or pythagorean fields? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the ...
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1answer
42 views

Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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1answer
73 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
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1answer
47 views

Easy explanation on primary decomposition of ideals. [duplicate]

The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes. Could someone explain to me how this can be done? Edited: My question ...
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1answer
740 views

What conditions guarantee that all maximal ideals have the same height?

It fails in general that all maximal ideals in a commutative ring with unity have the same height. It's easy to construct a counter-example when the ring is NOT an integral domain (consider the ...
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1answer
26 views

Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
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1answer
37 views

Flatness of quotient rings

The following is Exercise 2.4, in Chapter 1 of Liu, Algebraic Geometry and Arithmetic Curves: Let $I$ be a finitely generated ideal of $A$: $A/I$ is flat. $I^2 = I$. $I = (e)$ where $e^2=e$. I ...
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1answer
181 views

Is $R/N(R)$ a faithfully flat $R$-module?

I'm studying recently faithfully flat modules and I'd like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the ideal of ...
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2answers
316 views

$M\oplus A \cong A\oplus A$ implies $M\cong A$?

Let $A$ be a commutative unital ring and $M$ an $A$-module. Suppose that $M\oplus A \cong A\oplus A$. Then is $M\cong A$? We have that both $M\oplus A$ and $A\oplus A$ are biproduct for $(A, A)$ ...
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28 views

Local ring of an affine curve $K$ at a point $p\in K$

I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows: The local ring of $K$ at $P=(a,b)$ is the ring ...
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1answer
29 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
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1answer
190 views

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for ...
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1answer
254 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
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1answer
58 views

Ideal of 8 general points in $\mathbb{P}^2$

I am working through chapter 3 of Eisenbud's Geometry of Syzygies. In the first example he makes the claim that the ideal of 8 general points in $\mathbb{P}^2$ is generated by two cubics and a ...
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39 views

Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
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5answers
177 views

$K[[X]]$ is not a finitely generated $K[X]$-module.

How can I prove that $K[[X]]$ is not finitely generated over $K[X]$ as a module, where $K$ is a field. What I tried: if above is not true then $K[[X]]$ is integral extension over $K[X]$. But I ...
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1answer
21 views

A possible characterization of divisible modules

According to mathworld: Definition. Let $R$ denote a commutative ring and $M$ denote a module over $R$. Then $M$ is divisible iff for every $a \in R$, if $a$ is not a zero-divisor, then for all $x ...
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1answer
23 views

General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
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1answer
26 views

Is the localization of an injective cogenerator an injective cogenerator?

We know that in Noetherian rings any localization of an injective module is again an injective module. Is the localization of any injective cogenerator again injective cogenerator?
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1answer
92 views

Does such localization of integral extension preserve inclusion?

Let $R\subset T$ be two commutative rings, and $T$ is integral over $R$. Let $\mathfrak m\in \operatorname{Max} R,\mathfrak n\in\operatorname{Max}T$ such that $\mathfrak m=\mathfrak n\cap R$. Show ...
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2answers
78 views

Faithfully flat ring homomorphism properties

This is from Liu's Algebraic Geometry and Arithmetic Curves exercise 1.2.19 a. Let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that $f$ is injective and that $I\to I\otimes_AB$ ...
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56 views

Some questions about local rings and chain rings.

(All my rings are commutative with $1$.) The notion of a field spews forth many derived concepts. For example: An integral domain is a ring that can be homomorphically injected into a field. A ...
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1answer
17 views

Regular functions extension to normal points of varieties

I am doing the exercise 3.20 in Robin Hartshorne's Algebraic Geometry, Chapter 1. Let $Y$ be a variety of dimension $\geq2$, and let $P\in Y$ be a normal point. Let $f$ be a regular function on ...
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1answer
35 views

Example of a projective variety that is not projectively normal but normal

I want to prove the following statement: Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal. ...
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29 views

Localization and completion under a strong hypothesis

This question is closely related to this one, but in my case I think the hypotheses are different. Let $(A,\mathfrak m)$ be a regular, local noetherian domain (the local ring at a smooth point of ...
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0answers
33 views

Proof of Theorem 4.2.1 in Herzog-Hibi, “Monomial Ideals”

The Theorem and its proof can be found here. Specifically, i am stuck at the fourth paragraph of the proof. Let me give some context: Let $I$ be a graded ideal over a polynomial ring ...
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2answers
87 views

Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$ [closed]

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.
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142 views

Examples of rings whose polynomial rings have large dimension

If $A$ is a commutative ring with unity, then a fact proved in most commutative algebra textbooks is: $$\dim A + 1\leq\dim A[X] \leq 2\dim A + 1$$ Idea of proof: each prime of $A$ in a chain can ...
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1answer
448 views

Tensor products and polynomials with coefficients in a module

This is exercise $6$ in chapter $2$ on modules from Atiyah's and Macdonald's book. Let $M$ be an $A$-module and let $M[x]$ be the set of all polynomials in $x$ with coefficients in $M$. Then ...
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1answer
38 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
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1answer
31 views

Correspondence between prime ideals and irreducible algebraic sets

Let $k$ be an algebraic closed field. The Nullstellensatz theorem prove that $$I(V(J))=\sqrt{J}$$ and we have $$V(J)\text{ irreducible }\iff I(V(J)) \text{ prime }$$ So if $J$ is prime, $I(V(J))=J$ is ...
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1answer
76 views

Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
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1answer
27 views

Compute a Gröbner basis for $I=\langle f_1,f_2,f_3\rangle$.

Using lexicographic order compute a Gröbner basis for $$I=\langle f_1=xy^2-xy+y,f_2=xy-z^2,f_3=x-yz^4\rangle\subset \Bbb R[x,y,z]$$ I was strictly using these notes to compute a Gröbner ...
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1answer
48 views

Extension of DVRs and uniformizers

Let $(A,\mathfrak m)$ be a regular, Noetherian, local, domain of dimension $2$ and consider a prime ideal $\mathfrak p\subset A$ of height $1$. Moreover let $\hat{A}$ be the completion of $A$ with ...