Tagged Questions

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

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0
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1answer
25 views

Simple question on tensoring by a quotient ring

$A \subset B$ is an extension of commutative rings s.t. $B$ is a f.g. free $A$-module of rank $n$, so I have $A^n \stackrel{\sim}{\longrightarrow} B$ as $A$-modules. Let $\mathfrak a$ be an ideal of ...
3
votes
0answers
25 views

Contents of Tor modules

I'm interested in knowing a concrete description of what elements of Tor modules $\mathrm{Tor}^i_R(M,N)$ "are". As it stands I have no real intuition for, say, maps between Tor modules induced by ...
2
votes
1answer
14 views

Why does passing to the reduced ring not change the number of primes ideals?

I'm reading a note of Hochster's, and I don't follow something. He writes as the Corollary on page 9, Let $K\subseteq S$, where $K$ if a field, and $S$ is a finitely-dimensional $K$-vector space ...
0
votes
0answers
17 views

Question on Lemma preceding Going Up Theorem.

I have a question about Proposition 2.2.1 here: http://www.math.uiuc.edu/~r-ash/ComAlg/ComAlg2.pdf The proof has $S/R$ an integral extension of rings, $P_1,P_2$ prime ideals in $S$ lying over a prime ...
1
vote
3answers
54 views

Idempotents in $\mathbf{CRing}$

I'm not able to find an example of an idempotent morphism different from an identity in the category of commutative rings with unity (an idempotent, as a morphism in that category, must preserve 1, ...
-1
votes
1answer
47 views

Integral over a union of maximal two ideals

Let $A$ be Dedekind domain and $m_1$ and $m_2$ be maximal ideals of $A$ such that $A/m_1 \cong A/m_2$. How can I find a $x \in A-\{m_1 \cup m_2\}$ such that $x$ is not a root of any monic polynomial ...
2
votes
0answers
36 views

Eisenbud Corollary 6.7

Let $k$ be a field, $R=k[t]$ the polynomial ring in one variable, let $S$ be a Noetherian ring flat over $R$, If the fiber $S/tS$ over $t$ is a domain, and $U$ the set of elements of the form $1-ts$ ...
1
vote
1answer
29 views

on the statement of Theorem 3.3.7 in Bruns&Herzog

Let $\phi :(R,m) \rightarrow (S,n)$ be a local homomorphism of local Cohen-Macaulay rings, where $S$ is a finite $R$-module. In their proof of Theorem 3.3.7, Bruns&Herzog write that $\dim S = ...
4
votes
0answers
58 views

Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them are idealizations and valuation domains. But the first non-noetherian ring we are thinking about ...
3
votes
1answer
46 views

Tensor product and localisation

Let $k$ be an algebraically closed field and $K$ an extension field of $k$. Suppose $A$ is a finitely generated $k$-algebra which is a domain. Then we have a natural map $A \rightarrow A \otimes _ k ...
0
votes
1answer
17 views

showing that the Krull dimension of an extension module is zero

Let $(R,m,k)$ be a Cohen-Macaulay ring of dimension $d>0$ and let $M,C$ be CM $R$-modules such that $\dim M = 0, \dim C = d$. In the proof of Proposition 3.3.3-b(ii) in Bruns & Herzog, the ...
-1
votes
0answers
18 views

Integral elements and monic polynomials [duplicate]

Let $A \subset B$ be a ring extension, and let $f,g \in B[x]$ be monic polynomials such that $fg \in A[x]$. Is it true that the coefficients of $g$ and $f$ are integral over $A$? Please help me with ...
1
vote
1answer
46 views

Example of a module $M$ such that $\operatorname{depth}_{\mathfrak p}M<\operatorname{depth}_{A_{\mathfrak p}}M_{\mathfrak p}$; Matsumura, Ex. 16.5

I am looking for an example of a module $M$, a ring $A$, and a prime ideal $\mathfrak p$ such that $\operatorname{depth}_{\mathfrak p} M < \operatorname{depth}_{A_{\mathfrak p}} M_{\mathfrak ...
1
vote
1answer
46 views

Proof of the Auslander-Buchsbaum formula in Matsumura

There is a proof of Auslander-Buchsbaum formula in Matsumura's Commutative Ring Theory page 155. I am trying to understand the case $\operatorname{pd} M = 1$. He says take a short exact sequence $$ 0 ...
2
votes
1answer
46 views

Monic polynomials and integral elements.

Let $A \subset B$ be a ring extension, and let $f,g \in B[x]$ be monic polynomials such that $fg \in A[x]$. Prove that the coefficients of $f$ and $g$ are integral over $A$. My attempt was to prove ...
0
votes
1answer
16 views

Converse of the implication $V(S)\subseteq V(T)\iff T\subseteq\sqrt{\langle S\rangle}$.

I'm having trouble recalling one direction of the following bi-implication. Suppose $S,T$ are subsets of the polynomial ring $k[X_1,\dots,X_n]$ over an algebraically closed field. We have ...
3
votes
0answers
46 views

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for approaching ...
1
vote
1answer
37 views

Extending rings

This is a problem I've made up, which I cannot unfortunately solve. Any help will be appreciated. Let $R$ be a commutative ring with unity and $\operatorname{char} R=0$. I want to find the ring ...
1
vote
0answers
34 views

Elimination theory in Hartshorne

Does anyone know a good reference for elimination theory (Theorem 5.7A) mentioned in Hartshorne? The reference he gives is Van der Waerden modern algebra volume two, but it didn't feel locally ...
0
votes
2answers
96 views

If a module is nonzero, then a localization module is nonzero

Let $R$ be a commutative ring, when $\mathfrak p$ is a prime ideal, there is the localization $M_{\mathfrak p}:=S^{-1}M$, where $S=R\setminus\mathfrak p$. Show: If $M$ is a nonzero $R$-module, ...
0
votes
2answers
48 views

Non-zero prime ideals are maximal in the ring of algebraic integers

Let $A= \{y \in \mathbb{C} :$ $y$ integral over $\mathbb{Z}$ }. Let $P\not=\{0 \}$ be a prime ideal of $A$. I am supposed to prove that $P$ is also a maximal ideal. But I cant make it, is this really ...
0
votes
2answers
39 views

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$?

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$? Background: I was trying to prove that if $R{^m}$ is ...
0
votes
1answer
33 views

Contraction of non-zero prime ideals in the ring of algebraic integers

Let $A= \{y \in \mathbb{C} :$ $y$ integral over $\mathbb{Z}$ }. Let $P\not=\{0 \}$ be a prime ideal of $A$. Prove that $P \cap \mathbb{Z} \not= \{0 \}$. Iam totally stuck here, it is given that $P$ ...
1
vote
1answer
27 views

A property about quasi-primary modules

It is a fact that any discrete valuation domain $R$ has the property "P" that any proper submodule $N$ of any $R$-module $M$ is quasi-primary, in the sense that $\operatorname{rad}(N:M)$ is a prime ...
0
votes
0answers
44 views

Fibres of an ideal sheaf , total spaces and torsion groups

My question concerns a common example, which seems to often appear as an example/counter-example. Let $k$ be a field and consider the ideal exact sequence of the structure sheaf $k(p)$ of a point $p$ ...
0
votes
1answer
17 views

A doubt on a proposition involving Goldman domains.

$(*)$ Let $S/R$ be an extension of domains. Assume that for some $a\in R$, the ring $R[a]$ is Goldman. Then I want to show that $a$ is algebraic over $R$, whence $R$ is also a Goldman domain. DEF A ...
3
votes
1answer
80 views

Is the ring of entire functions coherent?

Call a commutative ring $R$ coherent if for each $n\in \{1,2,3,\cdots\}$ and each $n$-tuple $(r_1, ..., r_n)$ in $R^n$, the kernel of the map $R^n\owns (s_1, \cdots, s_n) \mapsto r_1 s_1 +\cdots + ...
3
votes
1answer
53 views

Is Spec ($k[x_1,x_2,\ldots])$ a smooth $k$-scheme?

Let $k$ be a field and let $A = k[x_1,x_2,\ldots]$. Note that $A$ is not of finite type over $k$. Is $\operatorname{Spec} A\to \operatorname{Spec} k$ a smooth morphism of schemes? I think it is ...
-1
votes
1answer
32 views

Graded ring localization. Why is this function bijective? [duplicate]

From Hartshorne, Chapter II.2, Proposition 2.5(b). If $R$ is a graded ring and $\mathfrak a$ is a homogenous ideal, then the function defined as $$\phi(\mathfrak a) = \mathfrak aR_f\cap R_{(f)}$$ ...
0
votes
2answers
134 views

Local ring and zero divisors

Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that all $x$ belonging to $\mathfrak m-\mathfrak m^2$ are zero divisors. Show that all elements of $\mathfrak m$ are zero divisors. My ...
0
votes
2answers
21 views

How to show that ideal is prime in $\mathbb{R}[x,y,z]$ modulo some other ideal

Let $R:=\mathbb{R}[x,y,z]$ and $g:=x^2+y^2-z^2\in R$. I would like to know how to show that $(x,y-z)/(g)$ is a prime ideal in $R/(g)$, and whether it is maximal or not. Thanks for the help!
-3
votes
0answers
110 views

Localization of modules and primary decomposition

Let $(R,\mathfrak m)$ be a Noetherian local ring and $M$ be an $R$-module. Suppose that for all prime ideals $\mathfrak p$, $\mathfrak p\ne\mathfrak m$, the localization $M_{\mathfrak p}$ is the ...
1
vote
2answers
42 views

Noether normalization for $k[x]_{x}$

According to the Noether normalization theorem, there exists a $k[t]$ where $t$ is an indeterminate and $k[t]\subseteq k[x]_{x}$ is a $k$-algebra extension so that $k[x]_{x}$ is a finitely generated ...
2
votes
1answer
40 views

Is the sum of saturated ideals saturated?

In a graded ring $S=\oplus_{k=0}^{\infty}S_k$, denote $m=\oplus_{k=1}^{\infty}S_k$, call an ideal $I$ to be saturated if $I=\cup_{n=1}^{\infty}(I\colon m^n)$. Is the sum of two saturated ideals still ...
1
vote
2answers
89 views

Torsion free module over a PID is flat

Suppose a ring of integers $S$ is an extension of a ring of integers $R$ with $\mathfrak{q}$ a prime ideal in $S$ and $\mathfrak{p}=\mathfrak{q}^c$ in $R$. Is there a straightforward way of showing ...
0
votes
0answers
24 views

Directs sum in exact sequences

I have a question about my homework question: Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting s.e.s. where $\alpha$ has a retraction $s$ and $\beta$ has a section $r$. ...
0
votes
1answer
32 views

Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields, suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
2
votes
1answer
49 views

The ideal for image of Segre embedding

How to show the ideal $(X_{ij}X_{kl}-X_{il}X_{kj})_{0\le i,k\le m, 0\le j,l\le n}\subset k[X_{ij}]_{0\le i\le m, 0\le j\le n}$ is radical? I can show the zero locus defined by the ideal is the image ...
0
votes
0answers
40 views

Generalization of Bezout Theorem to many-hypersurface case in Hartshorne's setting

I try to follow the ideas in Hartshorne's Chapter 1, Section 7. Suppose we have algebraic sets $Y_1,...,Y_l$, I try to define their intersection number $I(Y_1,...,Y_l)$ to be the leading term of the ...
1
vote
1answer
26 views

Localization of modules and minimal generating sets.

Let $A$ be a ring and $M$ a finite $A$-module; for $p \in \text{Spec} \space A$, write $\mathcal{K}(\mathfrak{p})$ for the residue field of $A_\mathfrak{p}$, and let $\mu (\mathfrak{p}, M)$ denote ...
-1
votes
1answer
54 views

Operations with ideals in a commutative ring

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in the ring $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?
1
vote
1answer
60 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
0
votes
0answers
19 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
5
votes
0answers
59 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
0
votes
4answers
116 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
0
votes
1answer
37 views

Showing local ring isomorphisms

This is a problem in K. Hulek's Elementary Algebraic Geometry. I figured out that $k[X]$ is the collection of polynomials of the form $f(x) + g(y)$ and also the local ring of an affine line at the ...
1
vote
0answers
20 views

GCD-Domain and proprieties

Let $A$ be a commutative GCD-domain (not necessary UFD or Bezout) and $a,b,c$ elements of $A$ such that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$. Is it true that $\gcd(ab,c) = 1$ ?
0
votes
1answer
36 views

An example of Noether normalization

Let $A=k[x_1,x_2]/(x_2^2-x_1^3+x_1)$. As an example of Noether normalization, determine elements $y_1,\ldots,y_m\in A$, algebraically independent over $k$, such that $A$ is a finite ...
1
vote
2answers
74 views

$k[x]/(x^n)$ module with finite free resolution is free

How to show a $k[x]/(x^n)$ module with finite free resolution is free? Suppose we have a exact sequence $k[x]/(x^n)^{\oplus n_1}\to k[x]/(x^n)^{\oplus n_{0}}\to M\to 0$, how do we get ...
0
votes
2answers
58 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...