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

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2
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1answer
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Existence of Hilbert's polynomial

I heard that Hilbert's syzygy theorem can be used to show the existence of Hilbert polynomials. How does the construction works? Namely, why do every coherent $O$-module $\mathscr F$ the ...
4
votes
1answer
83 views

Open Set of Non-zero Divisors of a Module

Let $R=k[x_1,\dots,x_r]$ be the polynomial ring over the field $k$. Denote by $R_1$ the vector space of linear forms, i.e. all the degree-$1$ elements of $R$. Let $M \neq 0$ be a finitely generated ...
1
vote
1answer
104 views

Hilbert Polynomial vs Hilbert Quasi-Polynomial

Let $R$ be an $\mathbb{N}$-graded ring with $R_0$ Artinian and $R = R_0[x_1,\dots,x_r]$, where the degree of $x_i$ is $d_i > 0$. Let $M$ a finitely generated $\mathbb{N}$-graded $R$-module with ...
3
votes
1answer
99 views

A non flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0$ for all $n\ge 1$

I want to find a non-flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0 \,\, \forall n\ge 1$, where $R=k[x,y]/(xy)$ and $k$ is field.
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votes
0answers
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Derivation (Matsumura: Commutative algebra)

I am reading Masumura, Commutative algebra, Chapter 10: Derivation. The following is in pages 177, 178. Two extensions $(C, \varepsilon, i)$ and $(C_1, \varepsilon_1, i_1)$ are said to be isomorphic ...
5
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1answer
130 views

Projective resolution of $k$ over $R=k[x,y]/(xy)$

I want to prove that $\operatorname{Tor}_{n}^{R}(k,k)=k\oplus k,\,\,\forall n\ge 1$. I found the projective resolution $$ R^4\stackrel{d_2} \longrightarrow R^3\stackrel{d_1} \longrightarrow ...
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vote
3answers
142 views

Question about localization

If $A \not= \{0 \}$ is a commutative ring and $P \subset Q$ are prime ideals of $A$ then of course $P \cap (A \setminus Q) = \varnothing$ so that $PA_Q = S^{-1}P$ is a prime ideal of $A_Q$ where $S=A ...
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1answer
47 views

An ordered group $G$ is Archimedean if and only if the following holds…

Let $G$ be an ordered group; then $G$ is Archimedean if and only if the following condition holds: $$\text{if} \space a, b \in G \space \text{with} \space a>0, \space \text{ there exists a ...
1
vote
1answer
76 views

Integral closure of a PID is torsion free

Can anyone explain me why the integral closure of a PID $A$ in a separable finite extension of its fraction field is a torsion free $A$-module? I know that it is a finitely generated A-module ...
2
votes
1answer
107 views

Equivalent conditions for an ideal to be prime

Let $R$ be a commutative ring. An ideal $I$ is called prime if whenever $ab\in I$ then $a\in I$ or $b\in I$. I want to show that $I$ is prime if whenever $JK\subseteq I$, then $J\subseteq I$ or ...
2
votes
1answer
109 views

Why are minimal irreducible closed sets in $A^n$ single points?

In Hartshorne's Algebraic Geometry example 1.4.4, he says A maximal ideal $m$ of $A = k[x_1,\cdots,x_n]$ corresponds to a minimal irreducible closed subset of $A^n$, which must be a point ... I ...
3
votes
1answer
138 views

Canonical Module and Socle of an Artinian $k$-Algebra

Let $R$ be an Artinian $k$-algebra generated by elements of degree $1$. Denote the canonical module of $R$ by $\omega_R$. By Theorem 3.6.19 in Bruns and Herzog (CMR), we have that $\omega_R = ...
1
vote
1answer
81 views

Symmetric algebra

If $V$ is a vector space over the field $K$ with basis ${v_1, v_2,…,v_n}$, then the symmetric algebra $S(V)= K[v_1,v_2,..,v_n]$. The question is: If $K$ is a commutative ring, then this equality is ...
4
votes
1answer
285 views

Local Noetherian domain of dimension one with principal maximal ideal

Let $(A,\mathfrak{m})$ be a local Noetherian domain of dimension one and suppose that $\mathfrak{m}$ is principal. I wish to show that every non-zero ideal of $A$ is a power of $\mathfrak{m}$. I have ...
6
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1answer
150 views

Local Artinian rings with a principal maximal ideal

I would be very grateful if someone would check my proof of the following result (this is not homework). All rings are commutative and unital. Proposition: If $(A,\mathfrak{m})$ is a local Artinian ...
3
votes
1answer
41 views

Why is $\hat{I}$ contained in the Jacobson radical $J(\hat{R})$?

Suppose $I$ is an ideal of a commutative ring $R$, and $\hat{R}$ is the $I$-adic completion. I don't follow why $\hat{I}$ is in $J(\hat{R})$. I know $\hat{R}$ is complete wrt the $\hat{I}$-adic ...
0
votes
1answer
100 views

Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
2
votes
1answer
53 views

Endomorphism ring as a set of matrices

Let $A=\mathbb Z[\sqrt{-5}]$, and let $I=(2,1+\sqrt{-5})$ (which is known to be a non-principal ideal of $A$ with $I^2=2A$). If we put $P=A \oplus I$, my question is: Why the endomorphism ring of ...
1
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1answer
129 views

judge if nilradical equals jacobson radical

judge if nilradical equals jacobson radical 1)a noetherian ring that is not a artin ring. 2)a local integral domain that is not a field. 3)a integral domain with only finite number of ...
3
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1answer
117 views

on the proof of Theorem 4.3.2 in Bruns & Herzog ``Cohen-Macaulay Rings" (Gotzmann's regularity theorem)

The theorem and the first part of its proof is shown below: In particular, the authors conclude (2 lines below equation (2)) that $(i): P_R(n) = {n + a_1 \choose a_1}+\cdots+ {n+a_r -(r-1) \choose ...
1
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1answer
101 views

Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
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votes
2answers
209 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
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0answers
56 views

Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
1
vote
1answer
59 views

Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
2
votes
2answers
62 views

MCS meet all prime ideals

let A be a commutative ring, is there any multiplicatively closed subset S (not containning 0), s.t. every prime ideal in A intercept S is not empty? My thinking is that there is 1-1 ...
5
votes
2answers
358 views

Where does the proof for commutative rings break down in the non-commutative ring when showing only two ideals implies the ring is a field?

We know in a commutative ring, if the only ideals are trivial and the whole ring, then the ring is a field, which is proved by every ideal is contained in a maximal ideal, which is proved by Zorn's ...
3
votes
1answer
125 views

Proof of Steinitz Theorem

I want a source containing the proof of Steinitz Isomorphism Theorem stating: For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$. Thanks!
3
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0answers
52 views

Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that ...
3
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1answer
82 views

Irreducible elements and unique factorization domain

Let $P=\{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. a) Which elements are irreducible in $P$: 4, 5, 6, 9, 10, 15? b) Find out, which one of rings: $ P$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
2
votes
1answer
104 views

If $P \in \operatorname{Supp}(M)$ prove that $P$ contains a prime ideal $Q$ with $Q \in \operatorname{Ass}_R(M)$.

My problem is below, Let $M$ be an $R$-module. The set of prime ideals $P$ of $R$ for which the localization $M_P$ is nonzero is called the support of $M$, denoted $\operatorname{Supp}(M)$. The set ...
5
votes
2answers
122 views

Units of $\overline{\mathbb{Z}}$

What are the units of $ \overline{\mathbb{Z}} $ (the ring of algebraic integers)? I know all roots of monic polynomials with constant term 1 are units, but are there any others?
2
votes
1answer
126 views

Regular Local Ring

Let $Y$ be an affine variety in $\mathbb{A}^n_k$ and $\mathfrak{i}$ its corresponding ideal. We use the notation $A(Y) = k[x_1,...,x_n]/\mathfrak{i}$ for the coordinate ring of $Y$. Pick a point $p\in ...
3
votes
2answers
91 views

Categorical Interpretation of Localization

At the very beginning of Ravi Vakil's amazingly famously amazing and famous notes on algebraic geometry, he remarks that some familiarity with localization and prime ideals is useful. I don't know ...
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1answer
122 views

A question on Artinian and Noetherian rings.

All rings are commutative and unital. Suppose that $A$ is a ring in which the zero ideal can be written as a product of maximal ideals of $A$. I try to prove that $A$ is Noetherian if and only if ...
7
votes
1answer
181 views

Is this particular module flat?

Let $A=k[x^2,xy,y^2]\hookrightarrow B=k[x,y]$, where $k$ is a field. Is $B$ flat over $A$? I am guessing the answer is no. My first thought is, since $B$ is integral over $A$, so it's finitely ...
7
votes
1answer
172 views

Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
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votes
2answers
99 views

The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
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1answer
56 views

$I$ and $J$ are coprime ideals iff $x \to (x + I, x + J)$ is surjective.

I'm stuck on this exercise and any help would be well appreciated: Let $R$ be a commutative ring with ideals $I,J$. Show that $R=I+J$ if and only if $\phi(x)= (x + I, x + J)$ is surjective from ...
4
votes
1answer
149 views

Castelnuovo-Mumford regularity of Cohen-Macaulay modules

Let $S=K[X_1,\ldots,X_n]$ and $M$ be a Cohen-Macaulay $S$-module. This equality holds $$ \operatorname{reg}(M)=\dim(M)+\max\{i\in\mathbb{Z}\colon P_{M}(i)\neq H(M,i)\}. $$ It's been proved in ...
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1answer
79 views

Picard group of $\mathbb Z[\sqrt{-5}]$

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of ...
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1answer
92 views

relation between units and non zero divisors in a ring

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?
3
votes
2answers
70 views

Can $ℂ$ be viewed as a (nontrivial) field of fractions?

Is there an interesting ring $S ⊂ ℂ$ such that $ℂ = Q(S)$? I’m thinking no, but how can I prove it?
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0answers
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Proving that a certain local ring is regular

I understand that this is a special case of the Jacobian criterion, but I was hoping that there was a simpler argument to prove it than for the criterion itself (I don't fully understand the proof of ...
4
votes
1answer
91 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
3
votes
1answer
86 views

Are finitely presentable modules closed under extensions?

If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable? The answer is "yes" if we replace modules with groups, ...
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1answer
146 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
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votes
3answers
291 views

Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement ...
2
votes
1answer
117 views

Describing $Spec(\mathcal{O}_K[X])$

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. I am trying to describe $Spec(\mathcal{O}_K[X])$ in terms of fibers of the map $g: Spec(\mathcal{O}_K[X]) \rightarrow ...
2
votes
1answer
62 views

prove that this ideal is radical

Let $A=\mathbb k[x,y,z]$ and let the ideal $$ I=(z-1,x^2-y).$$ I need to find $rad(I)$ but i don't know how. I think that this ideal is radical but I don't know good criteria for doing that =(
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vote
1answer
58 views

Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...