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

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1answer
101 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 $R$ ...
2
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1answer
64 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 ...
2
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0answers
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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0answers
84 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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0answers
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 $M$ ...
3
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0answers
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
321 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. ...
3
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1answer
216 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 ...
4
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1answer
125 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 ...
1
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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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2answers
76 views

$k[X]$ 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 ...
0
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1answer
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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0answers
98 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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1answer
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
73 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 ...
3
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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 ...
3
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0answers
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 ...
3
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1answer
194 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 ...
3
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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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2answers
155 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 / ...
1
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2answers
97 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, ...
3
votes
1answer
117 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 ...
3
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1answer
72 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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2answers
300 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 ...
2
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1answer
376 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 ...
1
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1answer
195 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 ...
1
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1answer
53 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 ...
2
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0answers
45 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 ...
3
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1answer
75 views

Wikipedia definition of an order (ring theory)

Wikipedia defines an order $\mathcal O$ of a finite type $\Bbb Q$-algebra $A$ to be a subring of $A$ satisfying the following properties. Here, by finite type $\Bbb Q$-algebra, I mean that $A=\Bbb ...
4
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0answers
113 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
5
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1answer
145 views

Why do people look into modules over Dedekind domains?

It is said in this blog that: The reason this turns out to be useful is that many examples in algebraic/arithmetic geometry require you to look no further than understanding modules over Dedekind ...
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0answers
56 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
6
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1answer
136 views

Local parameter of curves in affine n-space

I'm looking for a double answer to this question: a mathematical one (say, if the statement is correct or not) and a philosophical one (say, why we do expect this to be true, or not). Let $k$ be a ...
4
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0answers
75 views

Regular monomorphisms of commutative rings

What are the regular monomorphisms of $\mathsf{CRing}$? Is there a purely algebraic characterization? Since regular monomorphisms coincide here with effective monomorphisms (see Prop. 1. here), the ...
3
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1answer
79 views

tensor, symmetric, exterior power of a module over a PID

Let $R$ be a PID and $M\cong R^r\!\oplus\bigoplus_{i=1}^s\!R/Ra_i$. Denote the tensor, symmetric, exterior power of $M$ by $T^nM=\bigotimes_{k=1}^nM$ and $S^nM= T^nM/\langle ...
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2answers
375 views

A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
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2answers
112 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
5
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1answer
126 views

Proof that $K\otimes_F L$ is not noetherian

Let $F$ be a field and $K$ and $L$ be extension fields of $F$ such that $\mathrm{tr.deg}_F(K) = \infty$ and $\mathrm{tr.deg}_F(L) = \infty$. It seems to be proved that $K\otimes_F L$ is not ...
0
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1answer
74 views

Associated prime preserved under the quotient

Let $(R,m,k)$ be a complete local Noetherian ring and let $E$ be an $R$-module such that $\operatorname{Ass}E=\left\{m\right\}$. Let $N$ be a proper submodule of $E$. Question: Is it true that ...
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0answers
139 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
2
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1answer
155 views

Annihilators and exact sequences

Let $R$ be a commutative ring. Let $M_1$, $M_2$ and $M_3$ be $R$-modules. Let the following sequence be exact: $$0\longrightarrow M_1 ...
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1answer
82 views

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.r mean radical. I can't prove it,here is what I did. a is a intersection of prime ideals mean ...
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2answers
66 views

Can a chain of irreducible subvarieties always be extended to one of maximal length?

I'm interested in computing the dimension of a variety $X$. I can get a lower bound by exhibiting some strictly increasing chain of irreducible subvarieties $$\varnothing =Z_{-1}\subset Z_0\subset ...
8
votes
3answers
180 views

Why is the topology on $\operatorname{Proj} B$ induced from that on $\operatorname{Spec}(B)?$

In the proof of Lemma $3.36$ in Algebraic Geometry and Arithmetic Curves, it is stated that, if $B=\oplus_{d\ge0}B_d$ is a graded algebra over a ring $A,$ and if $I$ is an ideal of $B,$ then ...
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1answer
444 views

Difference between Matsumura's Commutative Algebra and Commutative Ring Theory

I am a beginner in more advanced algebra and my question is very simple, I would like to know the difference between these books of the same author, Hideyuki Matsumura Commutative Ring Theory ...
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2answers
122 views

$\mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$ and $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. We define similarly ...