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

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When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1$ and $I \subseteq R$ is called an ideal if it is a group with respect to addition and $rx, xs ...
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28 views

Field of fractions of $\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ [on hold]

Let $R=\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ and let $\mathbb{C}(X,Y)$ be the field of fractions of $\mathbb{C}[X,Y]$. Show that the field of fractions of $R$ can be expressed as ...
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1answer
25 views

If ideal can be generated by zero divisors, then is the depth of the ideal 0?

Let $R$ be a Noetherian ring and $I$ an $R$-ideal. The number $\operatorname{depth}_I R$ is the length of maximal $R$-regular sequence in $I$. It is well-known that If $\operatorname{depth}_I R = 0$, ...
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Does Magma let you specify primary invariants?

I am cross-posting this question from scicomp.SE. The computer algebra system Magma can calculate primary invariants (i.e. a homogeneous system of parameters) in an invariant ring of a finite group ...
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1answer
28 views

Example of an integral domain with a non-principal prime ideal of height one [on hold]

Is there an integral domain $R$ with a prime ideal $\mathfrak{p}$ of height $1$ which is not a principal ideal?
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22 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
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Are weakly étale ring homomorphisms of finite presentation étale?

Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat. Question. Are weakly étale ring ...
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45 views

Why is the map from $A^n$ to $M$ a surjective homomorphism?

How can one do the problem 1.3.11 b in Algebraic Geometry and Arithmetic Curves? I have read basics of commutative algebra but this one seems to be too difficult. Let $A$ be a commutative ring with ...
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1answer
49 views

Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
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2answers
51 views

Affine varieties and their ideals

I was reading on Wikipedia about quotient ideals. It mentions that if $W$ and $V$ are affine varieties (assume $V$ is) and $I(V)$ and $I(W)$ are the ideals for $V$ and $W$, then $$I(V):I(W) = ...
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1answer
25 views

$>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$

Let us define $R = k[x_1,\dots,x_t,x_{t+1},\dots,x_n]$; then it can be shown that $>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$ for all $1\leq i \leq t, t+1 \leq j \leq n$ ...
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1answer
38 views

Finite number of maximal ideals of bounded norm

Suppose that we have an integral extension of rings $R\subseteq S$ and $S$ is finitely generated as $R$-module or as $R$-algebra, and $R/\mathfrak m$ is finite for all maximal ideals and $S/\mathfrak ...
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1answer
38 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
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1answer
16 views

Poincare series and the Hilbert polynomial of $A = A_0[X_1,\dots , X_s]$

Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over an Artin ring $A_0$. This is a graded ring, and can be regarded as a graded module over itself. 1. What are the ...
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1answer
27 views

Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
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1answer
16 views

Poincare series and Hilbert polynomial of some graded modules

Let $k$ be a field, and let $k[X, Y ]$ be the polynomial ring in two variables equipped with the usual grading such that $\deg(X) = \deg(Y ) = 1$. Consider the ideals $I = (X, Y^2)$ and $J = (X^2, ...
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0answers
22 views

Minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$ [duplicate]

Write out a minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$, and determine the primes belonging to $I$. Determine the dimension of the ring $\mathbb C[X, Y, ...
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44 views

Elimination Ordering for the ring $k[x,y]$

How to show that the only elimination ordering on the ring $k[x,y]$ is the lexicographic ordering? (Ene and Herzog, Gröbner Bases in Commutative Algebra, Problem 3.1.) Definition (Elimination ...
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1answer
51 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
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A sufficient condition for factorization in a complete local ring

I think something like the following statement is true, but I don't recall a reference. Suppose $f(x,y)\in k[[x,y]]$ is power series with no constant or linear terms. Then, if the quadratic terms ...
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29 views

Nullstellensatz to prove Noether Normalization

In many commutative algebra texts, Noether Normalization Lemma is proved and then Hilbert's Nullstellensatz is obtained as a corollary. Nullstellensatz and Normalization Lemma seem to be non-trivial ...
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Proof that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$ [on hold]

I am trying to show that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$, where $\mathcal{P}$ is the polynomial ring $K[x_1, \dots, x_n]$ or ...
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1answer
28 views

$I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$

How to prove $I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$? Clearly $\sqrt{I(X_1)+I(X_2)} \subseteq I(X_1 \cap X_2)$ But for $f \in I(X_1 \cap X_2)$ $f(x)=0 \forall x\in X_1 \cap X_2$. how to show $f \in ...
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0answers
47 views

Spectrum and maximal spectrum of a ring

How do the $\mathrm{Spec}(\mathbb{C}\left [ X \right ])$ and $\text{m-Spec}(\mathbb{C}\left [ X \right ])$ look like? I understand the definitions of $\mathrm{Spec}(R)$ and $\text{m-Spec}(R)$ for a ...
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29 views

Buchberger's Algorithm Example

I've been reading Ideals, Varieties and Algorithms and came across an example of Buchberger's algorithm being computed and I am not able to understand how they came to have the final result. The ...
2
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1answer
47 views

Suppose A is a principal ideal domain with every ideal of finite index. Must A be a Euclidean domain?

Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain? If it's not known, are there any relevant partial results?
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1answer
48 views

Tensor product of Hom-module and another ring

Let $A$ be a local noetherian ring, $B$ and $C$ are finitely generated $A$-algebras and $M$ is a finitely generated $B$-module. Is the natural morphism $\mathrm{Hom}_B(M,B) \otimes_A C \to ...
6
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1answer
59 views

Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
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1answer
23 views

where do elements go under multiplication in a graded module?

Assume that $M = \bigoplus_{n = 0}^\infty M_n$ is a graded $A$-module, where $A = \bigoplus_{n = 0}^\infty A_n$ is a graded ring. We have by definition $A_m M_n \subset M_{m + n}$. Does this mean that ...
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18 views

Relation between minimal primes of a Noetherian graded ring and its subring

Let $A=⊕A_i$ be a Noetherian graded ring. Is there any relation between minimal primes of $A$ and minimal primes of $A_0$ (its $0$-th component)? In fact, my motivation is tight closure theory. I ...
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21 views

Reference request for numerical invariants of modules which are not finitely generated

Suppose that $R$ is an integral domain with subring $S$ and that both rings are finitely generated $k$-algebras ($k$ an algebraically closed field). $R$ is integral over $S$ if and only if $R$ is ...
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1answer
56 views
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Explanation of a proof about graded module structure

Let $\Bbb F$ be a field and $M$ a finitely generated $\Bbb F[x]$-module. The structure theorem for modules over a PID says that $$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb ...
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1answer
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how to prove an element is non-zero in a tensor-product

I was studying the following example from Atiyah & MacDonald's Introduction to Commutative Algebra: let $x$ be the non-zero element in $N := \mathbf{Z}/ 2\mathbf{Z}$, $M := \mathbf{Z}$, and $M' := ...
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1answer
28 views

Poincaré series pole at $1$

Let $A$ be a graded ring and $M$ a graded $A$-module. By $P(M,t)$ we denote the Poincaré series for $M$. In Atiyah and Macdonald, theorem 11.1 claims $P(M,t)=\dfrac{f(t)}{\prod _{i=1}^n ...
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The natural numbers form a distributive lattice under gcd and lcm. In arbitrary gcd domains, does gcd distribute over lcm?

Basically what it says in the title. If $A$ is a $\operatorname{gcd}$ domain, for any $x, y, z \in A$, does this identity hold? $$\operatorname{gcd}(x, \operatorname{lcm}(y,z)) = ...
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1answer
29 views

Does every non-archimedean absolute value on field take value in $\mathbb{Q}$

Let $K$ be a field, a non-archimedean absolute value is defined to be a map $K\to \mathbb{R}$ satisfying $|x|=0\Rightarrow x=0$, $|x|\cdot|y|=|xy|$ and $|x+y|\leq\max(|x|,|y|)$. Is there an example ...
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Origin of the name “Cauchy sequence” in abstract algebra

I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and ...
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1answer
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Integral element in $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Z}$ has form $m+n\sqrt{2}$ [duplicate]

I have studied about integral element. And I am not clear about an example. Could you please help me explain about it? Let R is a subring of S. $s \in S$ is integral if $s$ is a zero of some monic ...
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1answer
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What does the shifting of graded modules mean here?

For any graded module $M$ we denote $M(a)$ the module $M$ "shifted by $a$" so that $M(a)_d=M_{a+d}$. Thus for example the free $S$-module of rank $1$ generated by an element of degree $a$ is ...
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Geometric interpretation of multiplicity of an ideal.

Let $k$ be an algebraically closed field and let $R=k[x_1, \ldots, x_n]$. We write $\mathfrak{m}_p$ for the maximal ideal of $R$ corresponding to the point $p\in \mathbb{A}^n$. Given an ...
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1answer
24 views

Saturation of a multiplicatively closed subset

Exercise 3.7 of Atiyah-MacDonald asks the reader: if $A$ is a commutative ring and $\mathfrak{a} \triangleleft A$ an ideal, find the saturation of $1 + \mathfrak{a}$. Previously we have shown that ...
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Geometric intuition for normalization as intersection of valuation rings?

Why should the normalization of a ring correspond to the intersection of valuation rings containing it? I am looking for a geometric explanation, if possible. I understand that normalization at a ...
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1answer
44 views

Generators of the Tangent Space

Let $X$ be an affine variety, $X \subset A^n$ and suppose $f_1(T),\ldots,f_r(T) \in K[T_1,\ldots,T_n] $ generate $I(X)$. (Note that $I(X)$ is the ideal of $K[T_1,\ldots,T_n]$ of which elements of $X$ ...
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Two actions of rings in localization of modules

If $\mathfrak p,\mathfrak q$ are prime ideals of $R$. And $M$ is a $R$-module. Then $(M_\mathfrak p)_\mathfrak q$ is a $R_\mathfrak p$ and $R_\mathfrak q$-module under the action respectively 1) ...
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$f\in k(\mathbb{A^2})$ not regular at the origin implies it is not regular at points of a curve passing through the origin.

This is Exercise 4.12 (a) in Undergraduate Algebraic Geometry by Reid. Prove that any $f \in k(\mathbb{A}^2)$ which is not regular at the origin $(0, 0)$ also fails to be regular at points of a ...
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1answer
28 views

Associativity of tensor product over various rings

From Atiyah-MacDonald: Exercise 2.15. Let $A$, $B$ be rings, let $M$ be an $A$-module, $P$ a $B$-module and $N$ an $(A,B)$-bimodule (that is, $N$ is simultaneously an $A$-module and a $B$-module ...
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1answer
22 views

description of localization of pullback at a prime ideal

Can you please help me to know: How "the description of localization of $D$ at a prime ideal of it" is? Thank you.
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2answers
52 views

$\overline\phi: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.

Let $I$ be a nilpotent ideal in a commutative ring $R$, let $M$ and $N$ be $R$-modules and let $\phi : M \to N$ be an $R$-module homomorphism. Show that if the induced map $\overline\phi: M/IM \to ...
2
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1answer
20 views

Restricted valuation on subring of a DVR

Let $\mathcal O$ be a normalized discrete valuation ring. This means that there is a surjective valuation: $$v:\text{Frac}\,(\mathcal O)\rightarrow\mathbb Z\cup \left\{\infty\right\}.$$ Now consider ...
2
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1answer
46 views

Endomorphism of a finitely generated module and finite length of $\operatorname{Coker}f$

Let $M$ be a finitely generated module over a noetherian ring $R$ and suppose that $\dim R≤1$. And let be a one to one homomorphism $f:M\to M$. It is true that $\operatorname{Coker}f$ has finite ...