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

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4answers
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Find a “simpler description” for $\mathbb{Z}[X]/(X-5,X^2+3)$

The problem asks for a "simpler description" of the ring $\mathbb{Z}[X]/(X-5,X^2+3)$. I could use the Chinese Remainder Theorem if $\mathbb{Z}$ were replaced by $\mathbb{Q}$, but here the ideals ...
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3answers
37 views

Localisation commutes with taking quotients.

If $A$ is a ring, $S$ a multiplicative set and $I$ an ideal, write $T$ for the image of $S$ in $A / I$. Then $T^{-1}(A/I) \cong S^{-1}A/S^{-1}I$ and in particular, for a prime ideal $P$ we have that ...
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1answer
29 views

A construction in the Proof of Theorem 4.4.9 in Bruns&Herzog

Consider the following theorem and the part of its proof shown: So let $R$ be a homogeneous Cohen-Macaulay $k$-algebra with canonical module $\omega_R$. Let $b$ be the smallest degree for which ...
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0answers
53 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
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1answer
38 views

Tensor product of quotient and kernel

In my problem I have a PID $R$, elements $0\neq a,b\in R$ and a map $\phi_a:R\rightarrow R$ where $r\mapsto ar$. Assuming I have done all my previous calculations right I need to prove that ...
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1answer
44 views

Question about the support of a module $M$

Let $A \not= \{0 \}$ be a commutative ring and let $M$ be an $A$-module. Define $$\text{Supp} (M) = \{ P \in \operatorname{Spec} A : M_P \not= 0 \}$$ My first question is if $0$ is the element $0/1$? ...
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1answer
44 views

$k$-point after base change

If $X$ is a variety over $k$, is it true that there exists a finite separable extension $k'$ of $k$ such that $X$ has a $k'$-point? What if we can assume $X$ is a smooth projective curve? This seems ...
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1answer
68 views

Prove, that if the commutative ring has no zero divisors, then it is a field [duplicate]

Let $R$ be a commutative finite ring in which $ab = 0$ implies either $a = 0$ or $b = 0$ for any $a,b \in R$. Then, $R$ is a field. I do not understand how I should act. I tried different ways, but ...
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0answers
43 views

A minimal prime ideal consists of zerodivisors [duplicate]

Let $A$ be a unital commutative ring (I do not assume $A$ to be Noetherian). Let $\mathfrak{p} \subset A$ minimal prime ideal. Question: Are all elements of $\mathfrak{p}$ zero divisors? Comment: I ...
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1answer
66 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 ...
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1answer
21 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 ...
2
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1answer
60 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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0answers
39 views

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 ...
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1answer
52 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\longrightarrow R^3\longrightarrow R^2\longrightarrow R ...
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3answers
88 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
29 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 ...
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1answer
47 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 ...
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1answer
77 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 ...
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1answer
48 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 ...
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1answer
77 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 = ...
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1answer
45 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 ...
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1answer
53 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 ...
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0answers
43 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
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1answer
37 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 ...
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1answer
49 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
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0answers
43 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
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0answers
30 views

prime and maximal ideals in $\Bbb Z[x]$ [duplicate]

what are the prime ideals and maximal ideals in $\Bbb Z[x]$? I know $Z[x]$ is UFD but not PID, and (x) is prime but not maximal, and (x,2) is maximal. I wonder what should be the form of all ...
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0answers
25 views

shelling that facets are in order of non-decreasing dimension

A shellable simplicial complex is defined in Bruns-Herzog's book at Definition 5.1.11. It can be supposed that we define shellablity for non-pure simplicial complexes. so we can give examples of ...
2
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1answer
38 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 ...
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1answer
51 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 ...
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1answer
60 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 ...
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1answer
50 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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1answer
66 views

Why field of fractions of $k[x_1,x_2,…]$ is Noetherian? [closed]

the classical counterexample of a subring of a noetherian rings that is not noetherian is $k[x_1,x_2,...]$, which is not noetherian, but the field of fractions of $k[x_1,x_2,...]$ is, can anyone ...
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2answers
49 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
40 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 ...
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1answer
54 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 $t(sb'-s'b) = 0$. ...
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2answers
47 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
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2answers
219 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 ...
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1answer
32 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
35 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 ...
0
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0answers
66 views

immersions and finite morphisms

I have the following question: Let $X \subset \mathbb A^n$ be an affine variety. Prove that the immersion $i\colon X \hookrightarrow \mathbb A^n$ is a finite morphism. I know that the ...
3
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1answer
43 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$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
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1answer
69 views

Is ideal prime or maximal? [closed]

Find, whether or not given ideal of $\mathbb{Z}[x]$ ring is prime or maximal and describe the quotient ring : a) $J_1 = (x-5)$ b) $J_2 = (3, x+5)$. How can I do that?
2
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1answer
41 views

If $P \in Supp(M)$ prove that $P$ contains a prime ideal $Q$ with $Q \in Ass_R(M)$.

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 $Supp(M)$. The set of prime ideals ...
7
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2answers
86 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
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1answer
70 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
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2answers
63 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
49 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 ...
5
votes
1answer
148 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
80 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 ...