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

learn more… | top users | synonyms (1)

1
vote
0answers
45 views

Smith normal form Zariski-locally

Let $A\in GL_n(\mathbb{C}((t)))$, i.e. some invertible matrix over the ring of Laurent series. It is known that there are $P,Q\in GL_n(\mathbb{C}[[t]])$, such that $PAQ$ is diagonal. This is just ...
5
votes
2answers
315 views

Normal Ring and Prime Ideal whose Square is Principal

Let $k$ be a field of characteristic $\neq2$ and consider $R=k[x,y]/(y^2-x^3+x)$. Then (a) Show that $R$ is normal. (b) Let $P=(x,y)$ be a prime ideal of $R$. Show that $P^2$ is a principal ideal. ...
1
vote
0answers
39 views

Idempotents in a ring of fractions of the tensor product of Gaussian integers [duplicate]

Let $S=\{x^0,x^1,x^2,...\}\subset \mathbb{Z}$ be the multiplicatively closed subset generated by $x$. What are the nontrivial idempotents in the total quotient ring ...
1
vote
1answer
55 views

Localizing the quotient at a prime

I was reading in Dummit and Foote when I came across this statement I don't believe it is true. Let $R$ be a Dedekind domain (noetherian, integrally closed, domain of dimension 1). They say given ...
3
votes
2answers
404 views

The ring of convergent power series over $\mathbb C$ isn't noetherian

How can one prove that the ring of convergent everywhere power series in $\mathbb C[[z]]$ isn't Noetherian?
3
votes
1answer
114 views

On the Hilbert function of projective schemes

Let $X \subset \mathbb{P}^n$ be a projective subscheme (not necessarily reduced or irreducible). Denote by $I_X$ the ideal of $X$ i.e., $\Gamma_*(\mathcal{I}_X)$. There are two definitions of Hilbert ...
0
votes
2answers
61 views

Associated prime $\mathfrak{p}$ of $I$ written as $(I:x)$ for $x \notin \mathfrak{p}$.

Let $R$ be a Noetherian ring, let $I$ be an ideal of $R$ and let $\mathfrak{p}$ be a prime ideal associated to $I$. Then it is known from the standard theory that $\mathfrak{p} = (I:x)$ for some $x ...
0
votes
2answers
86 views

Nilradical of $\mathbb{R}[X,Y]/(X^nY^m)$

I'm having trouble finding the nilradical of $\mathbb{R}[X,Y]/(X^nY^m)$ for given $n$ and $m$. I believe the nilradical is $\{f(XY) \in \mathbb{R}[XY] : f \textrm{ has constant term 0}\}/(X^nY^m)\}$. ...
0
votes
2answers
221 views

Structure of maximal ideals of the quotient $\mathbb{C}[x,y,z]/ I$

I am trying to understand the general approach to the problems of the following type: Problem. a) Let $I\subset\mathbb{C}[x,y,z]$ be an ideal generated by $$\langle \ (x^2+y^2)^3+zx+3y^2z^3\ ,\ ...
1
vote
1answer
61 views

How to show that a ring is semilocal?

Let $R$ be a commutative, local ring and let $f$ be a monic polynomial in $R[x]$. How can I show that $R[x]/(f)$ is semilocal, respectively artinian? Thank you for your help!
0
votes
1answer
50 views

Equivalent definitions of algebra homomorphisms

I'm studying Atiyah-Macdonald's commutative algebra book and I'm trying to prove this equivalence: One implication If $h\circ f=g$. I can prove that $h(ax)=ah(x)$ but I have failed to prove that ...
6
votes
2answers
218 views

Idempotents in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$

Letting $\mathbb{Z}[i]=\left\{a+bi:a,b \in \mathbb{Z} \right\}$ be the ring of Gaussian integers, how many idempotents are there in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$? I came ...
0
votes
1answer
58 views

Torsion-free ideal

Let $I=(x^2,xy,y^2,p(z)x-q(z)y)$, where $K$ is a field and $p$ and $q$ relatively prime polynomials. How could I show that the ideal $(x,y)/I$ in the ring $K[x,y,z]/I$ is torsion-free as ...
2
votes
1answer
66 views

A question from Eisenbud, Commutative Algebra

On page 35, the proof of corollary 1.8: If k is an algebraically closed field and A is a k-algebra, then A = A(X) for some algebraic set X iff A is reduced and finitely generated as a k-algebra. In ...
2
votes
1answer
97 views

How to show that an extension is integral?

Let $R$ be a commutative ring and $I\subset R[x]$ an ideal in $R[x]$ that contains a monic polynomial. I want to show that $R/(R\cap I)\rightarrow R[x]/I$ is an integral extension. This is the ...
0
votes
1answer
93 views

If a union of ideals is closed under addition and multiplication, then all ideals are not prime

Let $J_1,\dots,J_n$, $n\geq 2$, be ideals of $A$, where $A$ is a commutative ring with unit. Suppose $X$ is a subset of $A$ closed under addition and multiplication, and $J_1,\dots, J_n$ is a minimal ...
1
vote
2answers
185 views

Invertible polynomials and power series [closed]

Consider the polynomial ring $A[x]$ and $f(x)=\sum_{i=0}^{n}a_ix^i\in A[x]$, where $A$ is a commutative ring with unit. Show that $f$ is a unit in $A[x]$ if and only if $a_0$ is a unit in $A$ and ...
5
votes
1answer
220 views

Pullback of maximal ideal in $k[y]$ is not maximal in $k[x]$.

Let $k$ be a field and let $k[x]=k[x_1,\ldots,k_m]$ and $k[y]=k[y_1,\ldots,y_n]$ be $k$-algebras. Let $\varphi:k[x]\to k[y]$ be a $k$-algebra homomorphism. It can be shown (pretty readily) that ...
2
votes
1answer
60 views

Tensor product of modules

Let $R$ be a polynomial ring over $\mathbb{C}$. Let $R_1=R/I$ for some ideal $I \subset R$. Let $M_1, M_2$ be $R_1$-modules. So, they are $R$-modules as well. Is it true that $M_1 \otimes_{R_1} M_2 ...
1
vote
1answer
155 views

Why is this ring not Cohen-Macaulay?

I am stuck with exercise 18.8, page 466 of Eisenbud's Commutative Algebra with a view towards Algebraic Geometry: Let $k$ be a field. The task is to prove that $R:=k[x^4, x^3y, xy^3, y^4] ...
4
votes
1answer
93 views

Are complete intersection prime ideals of regular rings regular ideals?

Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{p}$ be a prime ideal of $R$ which is a complete intersection, i.e. the minimal number of generators of $\mathfrak{p}$ equals its ...
0
votes
1answer
81 views

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 $k\in\mathbb{N}$, ...
5
votes
3answers
136 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 ...
1
vote
1answer
114 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
votes
4answers
113 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
votes
1answer
124 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) ...
0
votes
0answers
82 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
votes
1answer
367 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 $F,G$ ...
0
votes
1answer
31 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
45 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 ...
4
votes
2answers
149 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
42 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 ...
1
vote
0answers
60 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), ...
0
votes
2answers
97 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.
0
votes
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
votes
1answer
83 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.
11
votes
3answers
201 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
vote
1answer
50 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
votes
1answer
142 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
120 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
122 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
226 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
votes
2answers
200 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
vote
1answer
174 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
votes
1answer
104 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
95 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
85 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
votes
1answer
110 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
votes
1answer
105 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
227 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 $n$ ...