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

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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. ...
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1answer
145 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
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1answer
190 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 ...
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3answers
325 views

Example of a finitely generated flat module which is not free

I couldn't come up with an example of a finitely generated flat module which is not free. I know that over local rings, freeness and flatness are equivalent. So the ring cannot be a local ring.
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1answer
339 views

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
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1answer
123 views

Artinian affine $K$-algebra

Let $K$ be a field and $A$ an affine $K$-algebra. Show that $A$ has (Krull) dimension zero (is artinian) if and only if it is finite dimensional over $K$.
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294 views

Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$

I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z_1, y-z_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which ...
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186 views

Question on an isomorphism in the proof that $k[V \times W] \cong k[V] \otimes_k k[W]$

First I should say that I am aware of the existence of this question here and this question here. My question is a little different from these two because I am asking about a certain detail in the ...
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138 views

Can an ideal in a commutative integral domain be its own square?

If I is a non-zero proper ideal of a commutative integral domain, is it possible for I to be its own square?
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5answers
460 views

Localizations of Dedekind Domains are Discrete Valuation Rings

I am trying to prove the following implication, and can't seem to find my way around all the equivalent definitions of Dedekind domains and DVRs: I have a ring $R$ with the following properties: 1) ...
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1answer
356 views

Hilbert's Nullstellensatz without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. Can we prove the following theorem without Axiom of Choice? Theorem Let $A$ be a commutative ...
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1answer
381 views

System of generators of a homogenous ideal

Let $I$ be a homogenous ideal in the ring $k[x_{1},\dots,x_{n}]$. My question is: If $\lbrace f_{1},\dots,f_{r}\rbrace$ is a minimal system of generators of $I$, then are the integers $r$ and ...
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1answer
145 views

Dimension inequality for homomorphisms between noetherian local rings

$A$ and $B$ are commutative noetherian local rings with maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$ respectively. If $f\colon A \to B$ is a local ring homomorphism, how do I prove the ...
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1answer
136 views

Some question ideal of variety

For an affine variety $X=V(x^{2}+y^{2}-1, x-1)$, I found the ideal of $X$, $I(X)=\langle x-1,y\rangle$. But I don't know $I(X)=\langle x^{2}+y^{2}-1, x-1\rangle$.
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309 views

Prime spectrum and going-down property

I want to show that $f$ has the going-down property $\Leftrightarrow$ For any prime ideal $\mathfrak{q}$ of $B$, if $\mathfrak{p}=\mathfrak{q}^c$, then $f^{*}:\textrm{Spec}(B_{\mathfrak{q}}) ...
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0answers
77 views

Property of free submodules for a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
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69 views

Condition on a field that makes every subring an integrally closed domain

I want to know what condition would need to be additionally imposed on a field to make every subring of the field an integrally closed domain.
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1answer
119 views

Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $ \mathbb{C}^* $, and I'm confused. I don't have much background in this, so I was hoping a very concrete example would clear up a lot of this ...
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3answers
72 views

ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?

I am following an argument in chapter zero of Eisenbud's Commutative Algebra book. It is not clear whether or not he is assuming that $R$ is a domain. If I start the proof assuming $R$ is not ...
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1answer
198 views

showing Cohen-Macaulay property is preserved under a ring extension

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, with $R_0$ local Artinian. Assume also that $R$ is finitely generated over $R_0$ by elements of degree $1$. Let $M$ be a Cohen-Macaulay $R$-module. ...
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1answer
121 views

is the hilbert polynomial integer-valued everywhere?

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, finitely generated over $R_0$ with $R_0$ local Artinian. Let $M$ be a finite $R$-module of Krull dimension $d$. It is known that the Hilbert function ...
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2answers
147 views

On rings with a unique maximal ideal

I would be grateful if you guide me through the following question: Suppose a commutative ring with identity, $R$, has a unique maximal ideal, say $M$. If $M$ is principal, can we show that every ...
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2answers
143 views

Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
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2answers
124 views

Eisenbud Unmixedness Example

I am struggling with the following example in Chapter 18 of Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry , in which the author uses the unmixedness theorem to show that a genus ...
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2answers
179 views

Can $\operatorname{Spec}(R[X])$ ever be finite?

I have a quick question. Suppose $R$ is a nonzero commutative ring. Is it possible that $|\operatorname{Spec}(R[X])|<\infty$?
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408 views

Standard example where Jacobson radical not equal nilradical

Is there a standard example in commutative algebra of a ring where the jacobson radical does not equal the nilradical?
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1answer
191 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). ...
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1answer
472 views

A question on the symbolic powers of a prime ideal

In I. Swanson's notes about primary decomposition the author wrote: The smallest $P$-primary ideal containing $P^n$ is called the $n$th symbolic power of $P$, where $P$ here is a prime ideal of a ...
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1answer
77 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
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1answer
748 views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
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1answer
109 views

$S^{-1}B$ and $T^{-1}B$ isomorphic for $T=f(S)$

Let $f:A\to B$ be a homomorphism of rings, $S$ be a multiplicatively closed subset of $A$ and $T=f(S)$. Then $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules. First we define the ...
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102 views

Intersection of two localizations

Let $A$ be a commutative ring with unity. If $\mathfrak p,\mathfrak q\in \operatorname{Spec} (A)$ is it true the following equality $$A_\mathfrak p\cap A_\mathfrak q= A_{\mathfrak p\cup \mathfrak ...
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3answers
419 views

Points and maximal ideals in polynomial rings

Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
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2answers
160 views

$R$-linear injection [duplicate]

If $f: R^n\rightarrow R^m$ is an injective map, which is also $R$-linear, where $R$ is a commutative ring with unity. Is it true that $n$ has to be less than or equal to $m$ always?
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1answer
301 views

A formula for the minimum number of generators of a module over a semilocal ring

Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then $$\mu_R(M)=\max\{\dim_{R/\mathfrak ...
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3answers
666 views

Generators for the radical of an ideal

I am interested in finding a generating set of the radical of an ideal given a set of generators for the ideal itself, but after a lot of thought I cannot figure out a good way to do it. Specifically: ...
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1answer
463 views

A sufficient condition for a domain to be Dedekind?

We know that in a Dedekind domain, every nonzero ideal admits a unique factorization into a product of prime ideals. I was wondering if this condition is sufficient for a domain to be Dedekind, ...
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193 views

How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?

An ideal $M$ of a commutative ring $A$ (with unity) is maximal iff $A/M$ is a field. This is easy with the correspondence of ideals of $A/I$ with ideals of $A$ containing $I$, but how can you prove ...
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1answer
207 views

Verifying statement of exact sequences

I'm trying to read through Atiyah and MacDonald's Introduction to Commutative Algebra. Proposition 2.9 says a sequence of $A$-modules and homomorphisms $$ M'\stackrel{u}{\to} M\stackrel{v}{\to} ...
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2answers
166 views

Set of associated primes of direct sum

Let $M$ be a module over a ring $R$. Let $\operatorname{Ass}(M)$ be the set of annihilator ideals $\operatorname{Ann}(x)$, which are prime, so $$\operatorname{Ass}(M) = \{\operatorname{Ann}(x) \mid ...
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1answer
316 views

Computing with ideals: over $K$ or over $\mathbb{Q}\subseteq K$? does it matter?

I'm beginning to learn to use SINGULAR, the computer algebra system (CAS) for commutative algebra. NOTATION: If $K$ is a field of characteristic $0$, then $\mathbb{Q}\subseteq K$; otherwise ...
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459 views

Example of non-Noetherian non-UFD Krull domain?

After a confusing session of hopping through Wikipedia articles, I started trying to summarize for myself some of the inclusions and relations among the many types of integral domains. Right now I'm ...
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1answer
104 views

Atiyah-MacDonald, Exercise 5.4

I was having some trouble with the following exercise from Atiyah-MacDonald. Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Let $\mathfrak{n}$ be a maximal ideal of $B$ and let ...
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5answers
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Why does $p(a)=0$ imply $(x-a) \mid p$?

There's something I've never understood about polynomials. Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously, $$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$ The converse of ...
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122 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 ...
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100 views

Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
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2answers
161 views

How to show that $\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD [duplicate]

I am trying to show that $R=\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD. Let $I=(xw-zy)$. Let $X=x+I$, $Y=y+I$, $Z=z+I$, and $W=w+I$. My guess is that $X$ is irreducible and therefore $(X)$ is a ...
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2answers
121 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
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Why is the completion of the ring of germs of smooth functions $\cong \mathbb{R}[|T|]$?

Let $C^{\infty}$ be the canonical commutative ring on the set $\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ smooth}\}$. Let $\mathfrak{m}= \{ f \mid f(0)=0 \}$ a maximal ideal. Consider the ...
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An ideal whose radical is maximal is primary

I've got to prove that an ideal $Q$ whose radical is a maximal ideal is a primary ideal. That is, I want to prove that if $xy\in Q$, then $x\in Q$ or $y^n\in Q$ for some $n>0$. I've been ...