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

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5
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Rings in which every irreducible ideal is primary

Suppose $R$ is a commutative ring with $1$. It is well-known that if $R$ is Noetherian, then every irreducible ideal is primary (Lemma 7.12 in Atiyah & Macdonald). Is the converse true? That is: ...
0
votes
2answers
96 views

$Z(T)=Z(\mathfrak a)$, $\mathfrak a$ the ideal generated by $T$

I'm studying by myself the first chapter of Hartshorne's algebraic geometry as a introduction to this subject. I don't know how to prove this claim $Z(T)=Z(\mathfrak a)$. When we interpret the ...
1
vote
1answer
49 views

deducing inequality of polynomial degrees in a purely algebraic fashion

Let $k$ be field and $R=k[\xi_1,\cdots,\xi_r]$ be an integral domain of transcendence degree $t$ over $k$. Let $l(R_n)$ be the length of the $n^{th}$ homogeneous component of $R$. Then for $n$ large ...
6
votes
1answer
144 views

Are there Infinite Quotients of Algebraic Extensions of $\mathbb{Z}$?

It is well known that $\mathbb{Z}[a_1, \dots, a_n]/(a)$ is a finite ring if each $a_i$ is an algebraic integer and $a \neq 0.$ I suppose this statement becomes wrong if we just require those ...
2
votes
1answer
313 views

Local-global properties (localization): free, projective, injective, flat, torsion-free, etc?

Let $R$ be a commutative unital ring. We say that a property $(\ast)$ of modules is local-global when the following conditions are equivalent for any $R$-module $M$: $M$ is a $(\ast)$ $R$-module; ...
3
votes
3answers
1k views

If $R$ is an integral domain, then $R[[x]]$ is an integral domain

While solving another problem (specifically Exercise 7.2 in Atiyah & Macdonald's Introduction to Commutative Algebra), I got stuck in the following step: If $R$ is an integral domain, how I ...
5
votes
1answer
175 views

A commutative ring in which every prime ideal is 2-generated

Suppose $R$ is a commutative ring with 1. There are some statements that tells us if prime ideals behave in certain way, then all the ideals will behave in that way. For example, If every prime ...
2
votes
1answer
43 views

how different is the notion of an “indeterminate” from that of “algebraically independent” in relation to dimension theory?

The following is a well-known theorem in commutative algebra, see e.g. Matsumura, Commutative Ring Theory, p. 117: Let $A$ be a Noetherian ring and $X_1,\cdots,X_n$ indeterminates over $A$. Then ...
0
votes
1answer
75 views

Do we need Gröbner bases to study factor rings of polynomials?

I'm trying to understand how we can systematically study the factor rings of polynomials over a ring K. For example imagine that we're working in $R=K[x_1,...,x_n]$ and we have the ideal ...
5
votes
0answers
67 views

regular map of Noetherian rings

We say a homomorphism of Noetherian rings $\varphi:A\rightarrow B$ is regular if $\varphi$ is flat and for every prime ideal $p$ of $A$, the fiber ring $B\otimes_Ak(p)$ is geometrically regular over ...
8
votes
2answers
185 views

Reduced rings and tensor products

I assume all rings are commutative with identity. Denote $A':=A/ \sqrt{0}$ for convenience. Question is simple: For a given ring $R$ and $R$-algebras $A,B$, does this isomorphism $(A\otimes_R ...
5
votes
1answer
368 views

Isomorphism of formal power series factorrings over polynomials

This problem is taken from the Hartshorne's book Algebraic Geometry, Chapter 1, Section 5, Problem 14(a). Two polynomials $f(x,y)$ and $g(x,y)$ are written in the form $$f(x,y) = f_{r}(x,y) + ...
3
votes
1answer
64 views

proving isomorphism of two $k$-algebras

Let $k$ be a field. I would like to prove that $k[x,y]/(x^3-y^2) \cong k[t^2,t^3]$. Of course, intuitively, i can readily see that this must be the case. More formally, i define a homomorphism ...
2
votes
2answers
108 views

continuous images of Cauchy sequences in topological groups

on page 102 of Atiyah and MacDonald's "Introduction to Commutative Algebra", they state that if $G$ and $H$ are abelian topological groups and $f$ is a continuous homomorphism from $G$ to $H$, then ...
4
votes
1answer
88 views

writing the difference of two algebraic sets as an algebraic set

Let $S$ be an algebraic set of $k^n$, where $k$ is a field and let $f \in k[x] \doteq k[x_1,\cdots,x_n]$. I am interested in expressing $Y \doteq S-Z(f)$ as an algebraic set, i.e. as the zero set of ...
4
votes
1answer
164 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
2
votes
2answers
125 views

can we talk about zero sets of rational functions?

Usually, when talking about algebraic sets of the affine space $\mathbb{A}^n$, we refer to zero sets of collections of polynomials in $k[x_1,\cdots,x_n]$, where $k$ is the underlying field. ...
9
votes
1answer
309 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
4
votes
1answer
105 views

$k$-group endomorphisms of the multiplicative group scheme for $k$ a connected ring.

I wanted to verify that for a connected ring $k$ (i.e. a ring with connected spectrum, or equivalently without idempotents other than $0$ and $1$) the group of $k$-endomorphisms of the multiplicative ...
1
vote
1answer
115 views

a theorem on the dimension of finite algebras over a field (Hartshorne)

Robin Hartshorne in his Algebraic Geometry, Theorem 1.8A(b), p. 6, says that if $B$ is an integral domain which is a finitely generated $k$-algebra, $k$ a field, and $p$ a prime ideal of $B$, then ...
9
votes
1answer
189 views

Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!
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0answers
83 views

Non Classical Examples of Indecomposable Ideals

A classical example of a ring $R$ with an indecomposable ideal is the ring $C(X)$ of real valued continuous functions on $X$, where the $(0)$ ideal is not decomposable. Does anyone know other examples ...
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6answers
517 views

Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal

In a commutative ring with an identity, every maximal ideal is a prime ideal. However, if a commutative ring does not have an identity, I'm not sure this is true. I would like to know the ...
1
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1answer
124 views

Prime ideal whose square is not primary

Let $R = k[x,y,z]$ and $P = (y^2-xz,x^2y-z^2,x^3-yz)$ ideal of $R$. Show that $P$ is prime ideal. Show that $P^2$ is not primary ideal. Hint: Show that $(x,y,z)\in \operatorname{Ass}_R(R/P^2)$
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0answers
80 views

Systems of parameters for a $K$-algebra

I don't know how to solve the next problem: If we have two systems of parameters, $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ for a finitely generated $K$-algebra $A$ which is also an integral ...
2
votes
1answer
102 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
4
votes
1answer
57 views

Weak flat condition?

Let $R$ be a unit ring (not necessarily commutative). Then it is clear that for a right $R$-module $M$ we have: $M$ is flat $R$-module $\Rightarrow$ for any left $R$-module $E$ with $E\otimes_{R}M=0$ ...
6
votes
2answers
257 views

Extension of residue fields and algebraic independence

Let $A$ be a Noetherian integral domain, $B$ a ring extension of $A$ that is an integral domain, $P \in \operatorname{Spec} B, \, p = P \cap A$. Denote by $\kappa(p),\ \kappa(P)$ the residue fields of ...
14
votes
4answers
466 views

Isomorphism between quotient rings of $K[X,Y]$

Let $K$ be a field of characteristic $0$ and $m,n\in\mathbb Z$, $m,n\ge 1$. Prove that $$K[X,Y]/(X^2-Y^m)\simeq K[X,Y]/(X^2-Y^n)$$ if and only if $m=n$. (Related to Isomorphism between quotient rings ...
0
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1answer
104 views

Proof of Noether's Normalization theorem.

As stated here, Noether's Normalization Theorem states: Suppose that $R$ is a finitely generated integral domain over a field $K$. Then there exists an algebraically independent subset ...
5
votes
1answer
78 views

Finite dimensional commutative local algebras — reference request

What can be said about the structure of a finite dimensional, commutative, associative, unital local algebra over an algebraically closed field of characteristic zero?
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votes
2answers
259 views

Proof step in Steps in Commutative Algebra

I am trying to solve this exercise in Sharp's Steps in Commutative Algebra (I did not copy the question itself, just the first part of the exercise to prevent solutions of it): For me we should ...
3
votes
1answer
209 views

Proof details of Theorem 11.1 in Atiyah-Macdonald

I have some trouble filling in the details of this proof from Atiyah-Macdonald. In this result, the authors assume what follows: 1) $A = \oplus_{n=0}^\infty A_n$ is a Noetherian graded ring, and ...
13
votes
2answers
303 views

Example of two prime ideals whose intersection of the squares not equal to the square of the intersection

In this topic the OP raised the following question: Let $R$ be a commutative noetherian ring and $\mathfrak p,\mathfrak q \in \operatorname{Spec}(R)$. Is it true that $(\mathfrak p\cap \mathfrak ...
2
votes
0answers
105 views

Should I learn Commutative rings or finite fields “first” when self teaching?

My goal is to fully understand this answer on crypto.stackexchange by self-teaching myself all the basics. The term I'm working on now is a "Galois field", and starting on this wiki page. The header ...
1
vote
1answer
62 views

degree of remainder on division of multivariate polynomials

Let $f, g_1, \cdots, g_s \in \mathbb{R}[x_1,\cdots,x_n]$ and consider the division of $f$ by the $g_i$. Standard multivariate division algorithm will give $f = \sum_i a_i g_i + r$. I have been trying ...
7
votes
0answers
123 views

Let $A \subset B \subset C$ be subrings. If $C$ is finitely generated as an $A$-module, is $B$ finitely generated as an $A$-module? [duplicate]

Let $A \subset B \subset C$ be commutative rings. Suppose $C$ is a finitely generated $A$-module. Can we conclude that $B$ is a finitely generated $A$-module? Thoughts: Since $C$ is a finite ...
1
vote
1answer
95 views

Finding all $\mathbb{C}$-algebra homomorphisms $\mathbb{C}[x^2, y^2]\to\mathbb{C}$

I am interested in calculating all $\mathbb{C}$-algebra homomorphisms from $\mathbb{C}[x^2, y^2]\to\mathbb{C}$. I understand that every $\mathbb{C}$-algebra homomorphism will fix $\mathbb{C}$ and send ...
0
votes
1answer
64 views

Unclear step in Eisenbud's proof of a variant of Hauptidealsatz

I need help in understanding the proof of the following Theorem 10.1. If $\mathfrak{p}$ is a prime of a Noetherian ring $R$ minimal subject to containing $x \in R$, then its height is at most one. ...
3
votes
1answer
69 views

Algorithm for finding generators of an ideal

Let $k$ be a field, and $f:k[x_1,\ldots,x_n]\to k[y_1,\ldots,y_m]$ a $k$-algebra homomorphism. Given $r_1,\ldots,r_k\in k[y_1,\ldots,y_m]$, is there an algorithm for producing a finite generating set ...
2
votes
1answer
67 views

an algebraic set with Zariski tangent space equal to the entire ambient space at any point

Let $X$ be an algebraic set of $\mathbb{C}^n$ with vanishing ideal $I_X$. Let $p_1,\cdots,p_m \in \mathbb{C}[x_1,\cdots,x_n]$ be generators of $I_X$ and suppose that for any $i=1,\cdots,m$ and any ...
2
votes
1answer
116 views

for prime ideals, the intersection of the squares is the square of the intersection?

Here is something that i proved and i would appreciate feedback on my proof: Proposition: Let $A$ be a commutative Noetherian ring and $p,q \in \operatorname{Spec}(A)$. Then $p^2 \cap q^2 = (p\cap ...
4
votes
1answer
42 views

Fibres in a power series ring versus fibres in a polynomial ring (a simple question)

Let $A$ be a commutative ring and $p \in \operatorname{Spec} A$. In Matsumura's Commutative Ring Theory p. 118 it is mentioned that even though $A[x] \otimes \kappa(p) = \kappa(p)[x]$, it is not ...
1
vote
1answer
87 views

Regular local ring result - reference request

Reference needed for the following result: Let $R$ be a regular local ring with maximal ideal $\mathfrak m$. If $A$ is a flat $R$-algebra and $A/(\mathfrak m)$ is a domain, then $A$ is a domain. ...
2
votes
2answers
204 views

finitely generated & finitely related = finitely presented module?

Let $R$ be a ring $M$ an $R$-module. How can I prove that if $M\cong R^n/N$ for some $n\!\in\!\mathbb{N}$ and some submodule $N\leq R^n$ and if $M\cong R^{(I)}/\langle u_1,\ldots,u_m\rangle$ ...
3
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0answers
100 views

Castelnuovo-Mumford regularity

Let $R=K[x_1,\dots,x_{10}]$, where $K$ is a field. Consider $$I=(x_1x_7,x_1x_{10},x_2x_8,x_3x_9,x_4x_{10},x_1x_5x_9,x_2x_6x_{10},x_1x_4x_5x_8,x_2x_5x_6x_9,x_3x_6x_7x_{10})$$ which is a squarefree ...
2
votes
2answers
167 views

$A/ I \otimes_A A/J \cong A/(I+J)$

For commutative ring with unit $A$, ideals $I, J$ it holds $$A/ I \otimes_A A/J \cong A/(I+J).$$ A proof can be found here (Problem 10.4.16) for example. However, I'd be interested in a less ...
3
votes
2answers
85 views

do fibres of morphisms of Noetherian rings have finite Krull dimension?

Let $f:A \rightarrow B$ be a morphism of Noetherian rings. Let $p \in Spec(A)$ and let $C=B \otimes \kappa(p)$ be the fibre over $p$. Is it true that $\dim C < \infty$? How can we see that? ...
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vote
2answers
108 views

Proving that for certain ring of algebraic integers $R$, $R/bR$ is finite

This is a part of proof I try to understand. The situation is the following: Suppose that $a,b,x,y$ are algebraic integers such that $b \neq 0$ and $ax+by=1$. Set $K:=\mathbb{Q}(a,b,x,y)$ and ...
1
vote
2answers
127 views

How to calculate the embedding dimension of this ring

Define $$R=\frac{k[x,y,z]_{(x,y,z)}}{(x^2z,y^2z)}.$$ How can I prove that $\operatorname{edim}R-\dim R\leq1$ (where edim means minimal number of generators of the maximal ideal) ? It seems to me that ...