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

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2
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1answer
15 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
13 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 ...
4
votes
1answer
24 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
18 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
26 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
3answers
49 views

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
38 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
36 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
30 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 $P$ ...
0
votes
1answer
22 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
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0answers
37 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
16 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
70 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
50 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
27 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
114 views

Local ring and zero divisors

Let $(R,m)$ be a Noetherian local ring. Suppose that all $x$ belonging to $m-m^2$ are zero divisors. Show that all elements of $m$ are zero divisors. My progress: My idea is to see that the set ...
0
votes
2answers
20 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!
-1
votes
0answers
62 views

Localization modules and primary decomposition [on hold]

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
38 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
35 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 ...
2
votes
1answer
59 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
22 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
31 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
45 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
39 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
51 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 $AB$?
2
votes
1answer
56 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
56 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
111 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 ...
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0answers
18 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
33 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
70 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 ...
-1
votes
1answer
32 views

Surjective homomorphism of rings. Every ideal of B is an extended ideal of an ideal of A. [closed]

Let $f: A \rightarrow B$ be a surjective homomorphism of rings. I have to prove that every ideal of $B$ is an extended ideal of an ideal of $A$. Thanks! :)
0
votes
2answers
57 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 $I \subset \cup_i p_i$ ...
2
votes
1answer
52 views

Show structure of a commutative ring in a tensor product [on hold]

I need some help with this: Let $R$ be a commutative ring and $S$ and $T$ be commutative $R$-algebras. Show that $$ S \otimes T $$ has the structure of a commutative ring with multiplication: $$ (s ...
0
votes
2answers
31 views

Units and nilpotents in quotient ring. [closed]

$A$ is a commutative ring and $N(A)$ is the nilradical of $A$. If $A/N(A)$ is a field, show that every $a \in A$ is invertible or nilpotent.
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1answer
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Krull dimension of a finitely generated integral domain over $k$ is equal to the transcendence degree.

This theorem is from Matsumura (p.34) Let $k$ be a field and $A$ an integral domain which is finitely generated over $k$. Then $\dim A = \operatorname{trdeg}_k A$ (where $\operatorname{trdeg}_k ...
1
vote
1answer
15 views

About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
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vote
0answers
29 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
0
votes
1answer
19 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
0
votes
1answer
29 views

If $0\to M'\to M\to M''\to 0$ is exact, why does $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup \operatorname{Ass}(M'')$.

I'm stuck on a proof I'm reading. Let $0\to M'\stackrel{\mu}\to M\stackrel{\sigma}\to M''\to 0$ be a sequence of $A$-modules. Then $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup ...
1
vote
1answer
26 views

Factorization in Dedekind domains

Let $R$ be a commutative, Dedekind (and therefore Noetherian) ring with $1$. Let $I$ be a non-prime ideal of $R$, and let $a,b$ be elements of $R$ such that $a\not\in I,b\not\in I$ but $ab\in I$. Let ...
2
votes
2answers
76 views

Hartshorne II Prop 6.8

My weaknesses with commutative algebra are really slowing down my progress through Hartshorne. I hope someone can help me understand some statements in the proof of the proposition below. Prop ...
0
votes
2answers
23 views

If $\mathcal{I}(-)$ is the ideal map on subsets of affine space, why does $A\subseteq\overline{B}\iff\mathcal{I}(B)\subseteq\mathcal{I}(A)$?

I think this is a basic property of $\mathcal{I}(-)$, but I'm having trouble seeing it. I denote by $\mathbb{A}^n$ the affine $n$-space over an algebraically closed field $k$, where if ...
1
vote
2answers
55 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 - ...
2
votes
2answers
50 views

Show that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian

Prove that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian. I know that to be Noetherian, every ideal is generated by finitely many elements or equivalently R ...
2
votes
0answers
47 views

UFD and relatively prime elements

I've found the following statement at page 9 of Griffiths, Harris "Principles of Algebraic Geometry": Proposition. If $R$ is a UFD and $u$, $v \in R[t]$ are relatively prime, then there exist ...