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

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Is every module a direct limit of cyclic modules?

I want to show that $M$ is $A$-flat is equivalent to $Tor_1^A(M,A/I)=0$ for every finitely generated ideal $I$. I want to show $Tor^A_1(M,N)=0$ for any $A$-module $N$. Is every module a direct ...
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1answer
35 views

Homomorphisms from the base change of a module

Let $A, B$ be commutative rings with one and let $M$ be an $A$-module, $f: A \rightarrow B$ a ring homomorphism. Consider the (right) $B$-module $M \otimes_A B$. What can we say about ...
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Noetherian local ring and regular sequence

Let R be a commutative Noetherian local ring. If x is non zero-divisor in R, then {x} can be completed in a maximal regular sequence. someone can tell me if this is true? Thanks.
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1answer
45 views

Construct a counterexample of a primary ideal which …

Let $A$ be a Noetherian local ring of dimension $d$, $\mathfrak{m}$ its maximal ideal. Suppose $\mathfrak{q}=(x_1,\ldots,x_d)$ is an $\mathfrak{m}$-primary ideal. Suppose $f(t_1,\ldots,t_d)\in ...
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1answer
34 views

Cohen-Macaulay ring and saturated ideal

Let $A=\mathbb{C}[x_0,x_1,\dots,x_n]$ and I ideal of $A$. Is there any connection between "$A/I$: Cohen-Macaulay ring" and "$I$: saturated ideal"? Does one of them imply another? Please give me any ...
4
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1answer
50 views

Maximal ideals in the ring of eventually constant sequences of real numbers

For homework I am studying the ring $R$ of eventually constant sequences of real numbers (with multiplication and addition defined componentwise). What are the maximal ideals of $R$? By looking at ...
2
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3answers
105 views

What is $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$?

I would like to know what $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$ is isomorphic to, where $n,m\in\mathbb{N}$. Of course there will likely be cases depending on coprimeness and whatnot; ...
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1answer
26 views

Is there a topological characterisation of non-Archimedean local fields?

A local field is a locally compact field with a non-discrete topology. They classify as: Archimedean, Char=0 : The Real line, or the Complex plane Non-Archimedean, Char=0: Finite extensions of the ...
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0answers
48 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
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2answers
110 views

How can I complete the proof of Noetherianity of I. S. Cohen?

Theorem (I. S. Cohen). If $R$ is an unital commutative ring, and for each ideal prime $\mathfrak{p}\in Spec(R)$ we know $\mathfrak{p}$ is finitely-generated as $R$-mod then $R$ is Noetherian. ...
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1answer
28 views

If $M$ is a maximal ideal of a commutative ring R, then $R/M \cong R_M/M_M$. Where $R_M$ is the localization of $R$ at $M$.

If $M$ is a maximal ideal of a commutative ring R, then $R/M \cong R_M/M_M$. Where $R_M$ is the localization of $R$ at $M$. We know that localization commutes with taking quotients, so $R_M/M_M ...
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0answers
45 views

Why is this ring Cohen-Macaulay?

Let $q_1, q_2$ be quadratic homogeneous polynomials in $x_0,x_1,x_2,x_3,x_4$ over $\mathbb C$ and let $X_i:=V(q_i)=\{(a_0,\dots,a_4)\in \mathbb{P}_{\mathbb{C}}^4\mid q_i(a_0,\dots,a_4)=0\}$. If ...
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0answers
43 views

$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{Z}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$ [duplicate]

I want to prove that $$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{\mathbb{Z}}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$$ where $m ,n \in \mathbb{N}$ and $d = \gcd(m,n)$. Any ...
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0answers
102 views

Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
2
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1answer
39 views

Basic question in the radical of an ideal

Does the radical of an ideal $\sqrt{I}$ always contain $I$? I think the question boils down to if we are given $\sqrt{I}$, does it contain all the elements such that raised to ANY power? So $x \in ...
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1answer
36 views

Reference for $(N_1\cap N_2)\otimes_A M = (N_1\otimes_A M)\cap ( N_2\otimes_A M)$

Where can I find a canonical proof of the following statement? If $M$ is a flat $A$-module and $N$ is an $A$-module with submodules $N_1, N_2$, then $$(N_1\cap N_2)\otimes_A M = (N_1\otimes_A ...
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2answers
47 views

$\mathbb{Z}/n\mathbb{Z}$ is not flat

On the flat module Wikipedia page, it's stated that $\mathbb{Z}/n\mathbb{Z}$ is not flat over $\mathbb{Z}$. But I don't understand their explanation of why. It is said that ...
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1answer
50 views

Why are $(X_1), (X_1,X_2), \ldots$ prime ideals?

I was looking at the proof of the dimension of the polynomial ring $R[X_1,\ldots,X_n]$ and I had a question: Why are $(X_1), (X_1,X_2), (X_1,X_2,X_3),\ldots, (X_1,\ldots,X_n)$ prime ideals in this ...
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0answers
73 views

Ideal in $k[x,y]$ generated by two elements

Suppose $k$ is a finite field of order $q$. Let $f = \prod_{1 \leq i \leq s} (x + b_i y)$ and let $g = \prod_{1 \leq i \leq t} (x + c_i y)$, where $b_i, c_j \in k.$ I am interested in finding out ...
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38 views

Eisenbud 3.11(d) - A Uniform Bound on the Length of Certain Modules

I am trying to solve this exercise from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. There is a hint or possibly a solution in the back, but I want to try to get some more ...
3
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1answer
49 views

Localization of an integer quotient is a field

Let $R:=\mathbb{Z}/24\mathbb{Z}$ be our ring, $f: \mathbb{Z}\to R$ be the canonical quotient map (i.e. $f$ sends an element to its equivalence class) and $q$ be the ideal generated by $f(3 ...
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82 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
2
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1answer
58 views

A module over an algebra. Is it a vector space?

Let $A$ be an algebra over a field $k$. I would like to know if my understanding of the following correct or not. What I want to clarify is the definition of a module $M$ over $A$. I know the ...
5
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1answer
65 views

Height unmixed ideal and a non-zero divisor

Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal? Background: A ...
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2answers
27 views

How does one find a minimal primary decomposition?

What exactly does it mean for a primary decomposition to be "minimal" and is the a general method to obtain such decompositions? I've tried looking at some examples but they all give very little ...
2
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1answer
68 views

Let $(R,m)$ be * local and $R_m$ regular. Is R regular?

Let $(R,m)$ be *local and $R_m$ regular. Is $R$ regular?
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1answer
35 views

Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
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1answer
66 views

Showing that if $f,g \in k[x,y]$ are irreducible and not associates then $(f,g) \cap k[x] \ne 0$

There is a part of example 10.25.3 at http://stacks.math.columbia.edu/tag/00EX that I'm having trouble understanding. Here, $k$ is a field and $f,g \in k[x,y]$ are irreducible and are not associates. ...
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1answer
35 views

Localization of rings and integral closures

I looking at localizations of rings and I have the following problem: Let $R=k[x]$ be a polynomial ring ($k$ a field) and $R'=k[x^2])$ (note $R$ is integral over $R'$. Then if I look at the ideal ...
3
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1answer
59 views

Is there an example of commutative ring with exactly three prime ideals for which this property holds?

Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have ...
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1answer
32 views

existence of a finite-length maximal regular sequence

Theorem 16.7 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a Noetherian ring, $I$ an ideal of $A$ and $M$ a finite $A$-module such that $IM \neq M$; then the length of a maximal ...
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1answer
53 views

Formally smooth vs. smooth

A (commutative) algebra $A$ is called formally smooth if for any (commutative) algebra $R$ and an ideal $I\subset R$ such that $I^2=0$, any morphism $A\to R/I$ lifts to a morphism $A\to R$. Suppose ...
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1answer
53 views

Open set in the image of a dominant morphism of affine spaces

Let $k$ be an infinite field, $X=Y=\mathbb{A}^n_k$, and let $\varphi:X\longrightarrow Y$ be defined by $n$ algebraically independent polynomials. In particular, $\varphi$ is dominant (that is, ...
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1answer
32 views

Localization of a module is zero implies the multiplicatively closed subset contains a single element annihilating the module

I need to show that if $S$ is a multiplicatively closed subset of a ring $A$, $M$ is a finitely generated $A$-module, and $S^{-1}M = 0$, then there exists a single element $s$ in $S$ so that $sM = ...
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1answer
32 views

Spectrum of the ring $k[T]/(T^2)$

Consider the the ring $B = k[T]/(T^2)$ where $k$ is a field. If $I$ is a prime ideal in $B$ then $I = (a + bT)$ for some $a,b \in k$ (with $b \ne 0$). Then $T = a^{-1}T \cdot (a + bT) \in I$. Hence ...
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1answer
61 views

Calculating Grobner Bases

In this question, $ℚ[x,y,z]$ is endowed with the lexicographic order with $x > y > z$. Set $u:= x^2 + 2yz^2$ and $v:= y^2 - 3xz$. Denote by $J$ the ideal of $ℚ[x,y,z]$ generated by $u$ and $v$. ...
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1answer
73 views

$m$-primary ideal and $M\otimes_{A} A/m \neq 0$

Let $A$ be a commutative local ring with maximal ideal $m$. Let $M$ be a (not necessarily finitely generated) $A$-module. Let $x_{1},\dots,x_{n}$ be an $M$-regular sequence such that ...
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1answer
64 views

Local ring with principal maximal ideal

Let $R$ be a local ring such that the only maximal ideal $m$ is principal and $\bigcap_{n\in\mathbb{N}}m^{n}=\lbrace 0\rbrace$. I would like to prove that any ideal $I\neq\lbrace 0\rbrace$ of $R$ ...
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1answer
51 views

Need help for this proof in Matsumura's Commutative Ring Theory

I'm beginning to study Matsumura's Commutative Ring Theory and I'm trying to understand this theorem when $M$ is finitely generated: I have the following questions: First question: It seems ...
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33 views

When can a ring homomorphism to the integers modulo 2 be lifted to a homomorphism to the integers?

Let $A$ be a commutative ring with unity. Let $f: A \to {\mathbb{Z}}/2$ be a ring homomorphism to the integers modulo 2. When does there exist a lift $g: A \to {\mathbb{Z}}$ to the integers such that ...
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53 views

Isomorphism between quotient ring and its localization

Let $R$ be a domain, $P$ a prime ideal of $R$, and $k$ an positive integer. I am wondering if we have the isomorphism: $$ R/P^k\cong R_P/(PR_P)^k $$ where $R_P$ is the localization of $R$ at $P$. If ...
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1answer
87 views

When is $\operatorname{gr}_I (M)$ finite?

When is $\operatorname{gr}_I (R)$ (I mean associated graded ring of $I$) finite? When is $\operatorname{gr}_I (M)$ finite? ($M$ is an $R$-module.)
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Associated primes and finite base change

Let $R$ be an integrally closed commutative Noetherian integral domain. Let $R \subseteq S$ be a ring extension such that $S$ is also an integral domain and is finite as an $R$-module. Let $I$ be an ...
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1answer
45 views

completion of the canonical module

For a local Noetherian Cohen-Macaulay ring $(R,m,k)$ the canonical module is defined to be any maximal Cohen-Macaulay module of finite injective dimension and of type $1$. The canonical module is ...
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1answer
172 views

Homogeneous and maximal ideal in a $\mathbb Z$-graded ring

Is Exercise 2.8 from Marley's notes on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let $R$ be a graded ring and $M$ a homogeneous maximal ideal of $R$. Prove that $M =…⊕R_{-1}⨁m_0⨁R_1⨁…$, ...
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1answer
14 views

If $b+(a)$ is not a zero divisor in $R/(a)$, does it follow that $(a,b)=R$

Let $R$ be a commutative ring with identity. Let $a,b$ be elements of $R$. If $b+(a)$ is not a zero divisor in $R/(a)$, does it follow that $(a,b)=R$ ? The converse can be easily shown to be true. ...
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1answer
74 views

“M is reflexive” implies “M is MCM”. Is the converse true?

Let $(R,m)$ be a local integrally closed domain of dimension $2$. Let $M$ be a nonzero finitely generated $R$-module. We know that "$M$ is reflexive" implies "$M$ is MCM". Is the converse true? (By ...
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0answers
48 views

An easy infinite free resolution

I'm doing exercise 1.23 on Eisenbud's Commutative algebra, and I have the following situation: let $k$ be a field and $R = k[x]/(x^n)$. They ask for a free resolution of $R/(x^m)$, for some $m \leq ...
0
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1answer
67 views

The ideal $I=(3,2+\sqrt {-5})$ is a projective module

Let $R=\mathbb Z[\sqrt{-5}]$ and $I=(3,2+\sqrt {-5})$ be the ideal generated by $3$ and $2+\sqrt{-5}$. I'm trying to prove that $I$ is a projective $R$-module. I'm using the lifting property ...
2
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1answer
121 views

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? here $k$ is a field thanks for any help.