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

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1answer
52 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 ...
1
vote
1answer
59 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 $\phi(\lambda,p(...
4
votes
2answers
193 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 ...
2
votes
1answer
49 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 $K_A,...
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0answers
82 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
107 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 ...
3
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1answer
120 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.
14
votes
3answers
232 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 ...
1
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1answer
61 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 $k$-...
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1answer
154 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
votes
1answer
293 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
votes
1answer
179 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
votes
2answers
401 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 ...
4
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2answers
264 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 ...
1
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1answer
358 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
votes
1answer
117 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 why ...
2
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1answer
105 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
119 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 $\operatorname{codim}(R)=\...
5
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1answer
141 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
405 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 $n$ points? ...
3
votes
1answer
96 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 $k(Y)[X]...
7
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1answer
354 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
42 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
votes
2answers
91 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
126 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
vote
1answer
144 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 ...
0
votes
1answer
55 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 F_{1}(...
4
votes
0answers
137 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 ...
1
vote
1answer
166 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 $mx$...
2
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0answers
174 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
votes
1answer
140 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
votes
1answer
205 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: A\...
5
votes
1answer
773 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 ...
7
votes
1answer
156 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
votes
1answer
163 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 $...
6
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2answers
131 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
174 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
61 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
114 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
votes
0answers
83 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 ...
4
votes
1answer
1k views

Does quotient commute with localization?

Let $R$ be a commutative ring, and $I \subset R$ an ideal. If we choose an element $x \in R$ we can consider $(R/I)_x$ and $R_x/I_x$. In general, does localization commute with quotient? i.e. $(R/I)_x ...
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0answers
33 views

Submodules of a Noetherian module are finite intersections of irreducible submodules [duplicate]

If $M$ is a Noetherian $R$-module then every submodule of $M$ is a finite intersection of irreducible submodules. Please show me the way how to get the proof of this statement.
6
votes
3answers
406 views

Is $\mathbb{C}[[x]] \simeq \mathbb{C}[x]_{(x)}$?

Let $\mathbb{C}[x]$ be the ring of polynomials and $\mathbb{C}[[x]]$ the formal power series. Is $\mathbb{C}[[x]] \simeq \mathbb{C}[x]_{(x)}$? Is it true? Is there a geometric interpretation of this ...
0
votes
1answer
54 views

Knots and reducible spectra $\mathbb{C}[\![x,y]\!]/I$

Let $I=(y^2-x^3-x^2)$ be an ideal of $\mathbb{C}[x,y]$. I don't know why $\operatorname{Spec}(\mathbb{C}[x,y]/I)$ is irreducible but $\operatorname{Spec}(\mathbb{C}[\![x,y]\!]/I)$ is reducible. Do you ...
0
votes
1answer
187 views

$\operatorname{Spec}(R)$ not connected iff $R$ is a product of two rings [duplicate]

Let $R$ be a commutative ring. How can I prove that $\operatorname{Spec}(R)$ is not connected if and only if $R$ is isomorphic to the product of nonzero ring $A_1$ and $A_2$? If we consider $R=\...
6
votes
1answer
694 views

Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain

I want to show that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw)\subset\mathbb{A}^4$, is not a unique factorization domain. Morally, all we need to do is find some nonzero element that ...
4
votes
1answer
655 views

Working out the normalization of $\mathbb C[X,Y]/(X^2-Y^3)$

I'm trying to identify the normalization of the ring $A := \mathbb C[X,Y]/\langle X^2-Y^3 \rangle$ with something more concrete. First, $X^2-Y^3$ is irreducible in $\mathbb C[X,Y]$, making $\langle ...