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

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1answer
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Decomposable Tensors over Rings

Suppose $R$ is a commutative ring and $M$ is a $R$-module. Then we can define the tensor product $M\otimes_R M$ and more generally the $k$-fold tensor powers $\otimes_R^kM$ for any $k\in\mathbb{N}$, ...
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3answers
149 views

When does there exist a commutative ring $C$ that contains rings $A$ and $B$ as a subring?

The statement I'm trying to prove is the following: Let $A$ and $B$ be commutative rings, both of characteristic $0$. Then there exists a commutative ring $C$ that contains both $A$ and $B$ as ...
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1answer
137 views

Twisted Quartic Ideal

I was asked to do some computations involving a "twisted quartic" function $f : \mathbb{C} \to \mathbb{C}^4$ defined by $t \mapsto (t,t^2,t^3,t^4).$ However, I first know that I need to compute a ...
3
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4answers
124 views

$\mathbb Q+X\mathbb R[X]$ is not Noetherian

Let $A=\{q+r_1X+ \cdots +r_nX^n: q \in \mathbb{Q}, r_i \in \mathbb{R}\}$ be the polynomial ring with rational costant terms. I have to prove that $A$ isn't a noetherian ring. How can I prove it?
3
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1answer
162 views

A subring of polynomial ring with coefficients in a DVR that is not noetherian

Let $R$ be a discrete valuation ring, $K$ its field of fractions and $A=\{f\in K[T],f(0)\in R\}$. Let $\mathfrak{m}$ be the maximal ideal of $R$, $\mathfrak{m'}={\mathfrak{m}+KT+KT^2+\cdots}$. 1) ...
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0answers
84 views

Question about initial forms

I am working through Eisenbud's Commutative Algebra, and in Chapter $5$ he defines the following map. Say we have a filtration of modules ${\cal F}:M=M_0\supset M_1\supset\cdots$. Then for $f\in M$, ...
7
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1answer
389 views

Deligne's formula

Let $M$ be some $A$-module and $f \in A$. Why do we have an isomorphism $$\varinjlim_n \hom_A(f^n A,M) \cong M_f \text{ ?}$$ Background. Let $X$ be a scheme, $U$ an open subscheme, and $F,G$ ...
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1answer
36 views

software with a routine for the vanishing ideal of a finite set of points

I am looking for an algebraic software package that provides a routine that computes the vanishing ideal of a finite set of points. So far i am working with Macaulay2 but i have not been able to find ...
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1answer
48 views

Spectrum of $\mathbb{C}[x,y]^{\mathbb{C}^*}$

Let $\mathbb{C}[x,y]$ the ring of polynomials with $\mathbb{C}$-coefficients. We can define an action $\phi: \mathbb{C}^* \times \mathbb{C}[x,y] \rightarrow \mathbb{C}[x,y]$ such that ...
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2answers
171 views

Atiyah Macdonald Chapter 3 Problem 23 Part ii)

I am really confused about Atiyah Macdonald chapter 3 problem 23 part ii) The set up: Let $A$ be a ring and $X=\text{Spec}(A)$ be the set of prime ideals of $A$ with the Zariski topology. Let $U$ be ...
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1answer
44 views

determining an equality involving transcendence degrees of fields of fractions and residue fields

Let $(A,p)$ be a local integral domain and $B=A[x]$, where $x$ is an indeterminate. Let $P$ be a prime ideal of $B$ that contracts in $A$ to $p$, such that $\operatorname{ht}(P/pB)=1$. Denote by ...
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0answers
70 views

comparing transcendence degrees of field of fractions and residue fields

Let $A,B$ be integral domains such that $A \subset B$, $P$ a prime ideal of $B$ and $p$ its contraction in $A$. Let $K_A, K_B$ be the field of fractions of $A,B$ respectively and let $\kappa(p), ...
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2answers
102 views

Localization of $\mathbb Z/n\mathbb Z$ w.r.t. the set of all nonzero divisors

Let $R=\mathbb Z/n\mathbb Z$ and $S$ the set of all nonzero divisors of $R$. Then what is the localization $S^{-1}R$? Help me plz.
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1answer
68 views

A question on Modules

Let $M$ be a module over a ring $A$ and let $f_{1},...,f_{n}$ be elements of $A$ generating the unit ideal. Show that $M=0$ iff $M_{f_{i}}=0$ for $i=1,...,n$. I feel that this is closely related to ...
2
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1answer
105 views

Given a generating set $S$ of a free $A$-module $M$, must $S$ contain an $A$-basis for $M$?

Given a generating set $S$ of a free $A$-module $M$, must $S$ contain an $A$-basis for $M$? I'm not sure if this is true or not. I've tried using a Zorn's Lemma argument which failed.
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3answers
216 views

If $M\oplus M$ is free, is $M$ free?

If $M$ is a module over a commutative ring $R$ with $1$, does $M\oplus M$ free, imply $M$ is free? I thought this should be true but I can't remember why, and I haven't managed to come up with a ...
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1answer
55 views

A polynomial algebra that is free as an $A$-module

I'm working through some problems when I stumbled across a question asking about conditions for when the polynomial algebra $k[x_1,\ldots,x_n]$ is also a free $A$-module, where $A$ is some ...
0
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1answer
146 views

Monomial ideals: isomorphism problem for commutative algebras?

Let $I,J\unlhd K[x_1,\ldots,x_n]=K[x]$ be monomial ideals and $f\!: K[x]\to K[x]$ a graded isomorphism (given by a matrix $A=[\alpha_{i,j}]\in K^{n\times n}$, i.e. $x_i\mapsto\sum_j\alpha_{i,j}x_j$ is ...
0
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1answer
169 views

Why Spec R is quasi-compact?

I'm trying to understand this proof The only thing I didn't understand is why there exists a finite subset $L$ such that $1_R=\sum_{l\in L}i_l$. It should be a silly doubt, I'm sure I'm forgetting ...
0
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1answer
145 views

Finite dimensionality and maximal ideals

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Is the following equivalence true? A is finite-dimensional over $k$ if and only if $A$ has ...
2
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2answers
290 views

If $\{f_i\}$ generate the unit ideal in a ring, so do $\{f_i^N\}$ for any positive $N$ [duplicate]

Let $R$ be a commutative ring, and let $\{f_i\}$ be a finite set of elements generating the unit ideal in R. Then $\{f_i^N\}$ also generate the unit ideal in $R$, for any positive $N$. Why is this ...
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2answers
226 views

About the injection $M \hookrightarrow \mathbb Q \otimes_{\mathbb Z} M$.

I want to prove that every abelian group can be embedded in a divisible abelian group. So I tried $M \rightarrow \mathbb Q \otimes_{\mathbb Z} M, m \mapsto 1 \otimes m$. It is obvious that $\mathbb Q ...
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1answer
227 views

The intersection of two minimal prime ideals.

Let $A$ be a reduced commutative ring (that is, $A$ has no nontrivial nilpotents) and $P_1$, $P_2$ two minimal prime ideals of $A$. Is it true that the intersection of $P_1$ and $P_2$ is zero? It ...
0
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1answer
107 views

When $Rx = Re$ and $e^2 =e$

Let $R$ be a commutative ring with identity. Suppose $x , e \in R$ with $Rx = Re \mbox{ and } e^2 = e$. what is the best thing that we can say about $x$?
4
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1answer
111 views

Is injectivity of algebras preserved by tensor products?

Suppose $R' \subset R$, $S'\subset S$ are inclusion of $k$-algebras. Does it hold that $R'\otimes_kS' \rightarrow R \otimes_k S$ is injective ? I know there're counterexamples for modules, but ...
2
votes
1answer
91 views

dimension of a projective variety

Let $Y$ be a projective variety with homogeneous coordinate ring $S(Y)$, where $S=k[x_{0},x_{1},\cdots ,x_{n}]$ and $k$ is algebraically closed. Show that dim $S(Y)=\text{dim} Y+1$. $$\text{My ...
2
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1answer
116 views

On the multiplicity of complete intersections

Suppose $R$ is a complete intersection. How can I prove that $\operatorname{mult}(R)\geq2^{\operatorname{codim}(R)}$, where $\operatorname{mult}(R)$ is the multiplicity and ...
4
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1answer
115 views

PID with infinitely many maximal ideals, irreducible, generic points.

I am trying to do this question and will appreciate if anyone gives comment on my attempt. I am sure there are mistakes somewhere, so I will be glad if someone points them out to me: Let $A$ be a ...
6
votes
2answers
308 views

Spec of tensor product of fields

Suppose $K/k$ is a finite separable extension of degree $n$. How to show that there exists a finite separable extension $k'/k$ such that $\operatorname{Spec}(K \otimes_k k') $ consists of $n$ ...
3
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1answer
86 views

About the irreducibility in $k[X,Y]$ and in $k(Y)[X]$

Let $k$ be a generic field and $k(Y)$ be the field of rational function in the variable $Y$. If $f\in k[X,Y]$ is an irreducible polynomial, is it true that it is irreducible as polynomial in ...
5
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1answer
293 views

Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
2
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0answers
39 views

Does $M$ finitely presented and $N$ finitely generated imply Hom$_R(M,N)$ f.g. when $R$ is not Noetherian? [duplicate]

If $R$ is a non-Noetherian ring, $M$ is a finitely presented $R$-module, and $N$ is a finitely generated $R$-module, does it hold that Hom$_R(M,N)$ is a finitely generated $R$-module? We tried ...
0
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2answers
89 views

What are the closed subsets of $\operatorname{Spec}(\mathbb{Z})$?

I'm trying to find what the closed subsets of $\operatorname{Spec}(\mathbb{Z})$ are. I know that the prime ideals of $\mathbb Z$ are the ideals generated by prime numbers, i.e., the prime ideals of ...
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2answers
108 views

Can we find a subset of $Spec(R)$ not quasi-compact?

If $R$ is a commutative ring with unit, we can easy prove that $Spec(R)$ is quasi-compact. However can you give me an example of $R$ such that a subset $A \subset Spec(R)$ isn't quasi-compact?
1
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1answer
133 views

proving that $\dim A[X] = \dim A + 1$ (Matsumura)

Let $A$ be a Noetherian ring and $X$ an indeterminate over $A$. I am having trouble understanding Matsumura's proof (Commutative Ring Theory, Theorem 15.4) that $\dim A[X] = \dim A + 1$. Below, i ...
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1answer
50 views

A completely reducible module is isomorphic to its associated graded module?

If $F.(M)$ is a (finite) filtration of a finitely generated module $M$ that is completely reducible, then $M \cong \operatorname{gr}_{F.(M)}$? Let $0=F_{n+1}(M) \leq F_{n}(M) \leq \cdots \leq ...
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0answers
100 views

Module of smooth vector fields

I want to show that the module of smooth vector fields is a free module over the ring of infinitely differentiable functions on some open subset of Euclidean space. I understand how to prove this from ...
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1answer
152 views

A question on the Chinese Remainder Theorem

This is a question from Lang's ANT, Thm 2 (ch.7, $\S2$). Let $k$ be a number field and $A$ its adele group. In the proof, Lang states Given $x\in A$, let $m$ be a rational integer such that ...
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0answers
130 views

Irreducible polynomials as formal power series

I'm studing the ring of formal series with complex coefficients $\mathbb{C}[[x]]$. I proved that the polynomial $y^2-x^3-x^2$ is irreducible in $\mathbb{C}[x,y]$ but reducible in $\mathbb{C}[[x,y]]$. ...
2
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1answer
128 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of ...
2
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1answer
173 views

augmented algebras and their morphisms

Let $R$ be a commutative unital ring and $A$ an associative (unital) $R$-algebra. What is an augmented $R$-algebra? A (unital) $R$-algebra $A$, together with a (unital) ring morphism $\varepsilon: ...
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1answer
503 views

Does Localization Commute with Direct/Inverse Limits

Let $A$ be a ring and let $M_n$ be $A$-modules. For a prime ideal $P$ in $A$ is it true that $$(\varprojlim_n M_n)_P=\varprojlim_n (M_n)_P\text{ and } (\varinjlim_n M_n)_P=\varinjlim_n (M_n)_P?$$ If ...
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1answer
149 views

Derived category of certain ring

I'm interested in the structure of $D^b(R)$, where $R=k[x]/(x^n)$. How one can describe this category? What is the list of indecomposable objects in this category?
6
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1answer
155 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
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1answer
55 views

A set of prime factors of an integer in $\mathcal{O}_k$

I've got a basic question from Thm 2 (ch.7, $\S2$) of Lang's Algebraic Number Theory. Let $k$ be a number field and $A$ its adele group. Let $S_{\infty}$ be the set of Archimedean absolute values of ...
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2answers
119 views

Concept of a subring in Atiyah-Macdonald's book

I think this definition is wrong, because nothing guarantees that the subring is closed to additive inverses. Thanks
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1answer
151 views

If for all $r\in R$ the element $ar+1$ is invertible in $R$, then $a$ belongs to the Jacobson radical

Let $R$ be a commutative ring with unity, and let $a$ be a fixed element of $ R $. Suppose that for every $ r \in R $, $ ar + 1 $ is invertible in $ R $. Show that $ a $ belongs to the Jacobson ...
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0answers
59 views

Are there in $(\mathbb{C}[x,y,z]/(x^3+y^3+z^3))_{x}$ exactly $12$ lines?

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ be the coordinate ring of the affine variety defined by the equation $x^3+y^3+z^3=0$. We can consider the localization in the element $x$, denoted by $R_x$. I ...
3
votes
2answers
112 views

(Integer) Variant of Hilbert’s irreducibility theorem

Let $P\in{\mathbb Q}[X,Y]$ such that $P(x,.)$ has an integer root for any integer $x\in{\mathbb Z}$. Does it follow that $P$ has factors of the form $Y-Q(X)$ for some $Q\in{\mathbb Q}[X]$, and does ...
2
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0answers
77 views

How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...