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

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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 ...
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1answer
76 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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169 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.
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1answer
417 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 ...
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1answer
188 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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67 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 ...
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72 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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43 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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95 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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58 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 ...
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273 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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47 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
118 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
88 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
126 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
47 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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125 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
95 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
93 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
63 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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55 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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83 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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133 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? if 1) $R$ is non-Noetherian ring , 2) $R$ is Noetherian ring and $M$ ...
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100 views

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
54 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
313 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
17 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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213 views

“M is reflexive” implies “M is maximal Cohen-Macaulay”. Is the converse true?

Let $R$ 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 maximal Cohen-Macaulay". Is the ...
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1answer
120 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 ...
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1answer
105 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 ...
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1answer
200 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.
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$k[X]$ is integral over $k[X^{2}]$

I am trying to show that $k[X]$ is integral over $k[X^2]$, where $k$ is a field. Taking an element $b=b_nx^n+b_{n-1}x^{n-1}+...b_1x+b_0 \in K[X]$ we want to find $a_i \in K[X^2]$ such that ...
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34 views

$(x_1,\ldots x_n)=(1)\implies (x_1^{k_1},\ldots, x^{k_n})=(1)$ [duplicate]

Let $R$ be a commutative ring with unit, I'm trying to prove why in this ring $$(x_1,\ldots x_n)=(1)\implies (x_1^{k_1},\ldots, x^{k_n})=(1)$$ It seems an easy question, but I couldn't prove it, I ...
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95 views

Jacobson Radical

Let $R$ be a local ring with unity then when can we say that the radical of Jacobson of $R, J$, is a $R/J$ module. By local I meant it has a unique ideal maximal. And when is $R$ is isomorphic to ...
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42 views

$f:M\rightarrow N$ module homomorphism, $(N/\mathrm{Im}f)_m=N_m/\mathrm{Im}f_m$

$f:M\rightarrow N$ is an $R$-module homomorphism and $f_\mathfrak{m}:M_\mathfrak{m}\rightarrow N_\mathfrak{m}$ is the induced $R$-module homomorphism $$f_\mathfrak{m}(m/s)=f(m)/s$$ where ...
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1answer
71 views

Depth of infinite direct sum

Let $R$ is a local ring, from the depth lemma, we can get $\operatorname{depth}(R\oplus\dotsb\oplus R)=\operatorname{depth}(R)$, here the direct sum is finite, how about the infinite case? By the ...
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1answer
58 views

Nilpotency of finite ideal

Suppose we have a commutative local ring $R$ with unit. I'm curious about whether the following statements are correct: 1- every proper finite ideal is nilpotent. 2-every proper finitely generated ...
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152 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
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187 views

Monic irreducible polynomial over an integral domain

These days, I have some basic problem in abstract algebra. I know that in any integral domain, any prime element must be an irreducible element. Moreover, if $A$ is a UFD, then an element $a \in A$ is ...
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1answer
56 views

Computing the closed subschemes of the projective line over a field

(Specifically, this is III-15 in E&H, but I feel like I've hit a brick wall in actually applying the definitions they've given to this example.) In Chapter I of The Geometry of Schemes, E&H ...
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1answer
35 views

Question about proof that every f.g. projective module over a local ring is free.

I'm reading the proof here. I'm at the line where they say $$ \psi\pi(f)=\psi(f+FR)=\varphi(f)+PR.$$ Since $\psi\pi$ is surjective, it should follow that $\{\varphi(f)+PR:f\in F\}=P/PR$. I don't ...
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154 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
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2answers
93 views

Existence of module of finite injective dimension

At p. 107 of the book Cohen-Macaulay Rings by Bruns and Herzog, the authors write "any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, ...
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114 views

Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
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1answer
71 views

Galois cover an affine scheme

Let $X = \operatorname{Spec}(A)$ be an affine scheme, with $A$ noetherian (and normal if this is useful). We suppose that $X$ is a finite étale covering of $Y = \operatorname{Spec}(B)$, Galois with ...
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291 views

When is a local, reduced, (commutative) ring an integral domain?

Question I am wondering whether or not it is true that if $A$ is a reduced ring, then is it the case that the localization of $A$ at any of its prime ideals is an integral domain? Discussion ...
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1answer
357 views

Is any UFD also a PID?

Is there any counterexample that will disprove that every unique factorization domain (UFD) is also a principal ideal domain (PID)? I mean, any PID is a UFD, does the converse hold? Thanks in ...
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1answer
194 views

How many ways are there to represent a monomial order, defined by $>$, by term order via matrices?

During the lecture, my professor brought up the list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the ...
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1answer
52 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
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44 views

Under what conditions are the resolutions of two modules subcomplexes of the resolution of the tensor product?

I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing ...