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

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4
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1answer
120 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 ...
4
votes
2answers
143 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 ...
4
votes
2answers
140 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 ...
4
votes
2answers
123 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 ...
4
votes
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 $|\operatorname{Spec}(R[X])|<\infty$?
4
votes
2answers
389 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?
4
votes
1answer
188 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). ...
4
votes
1answer
434 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 ...
4
votes
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 ...
4
votes
2answers
98 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 ...
4
votes
3answers
398 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 ...
4
votes
2answers
159 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?
4
votes
1answer
300 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 ...
4
votes
3answers
633 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: ...
4
votes
1answer
452 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, ...
4
votes
4answers
190 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 ...
4
votes
1answer
201 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} ...
4
votes
2answers
159 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 ...
4
votes
1answer
315 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 ...
3
votes
5answers
150 views

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 ...
3
votes
3answers
121 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 ...
3
votes
2answers
99 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 ...
3
votes
2answers
144 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 ...
3
votes
2answers
120 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 ...
3
votes
0answers
75 views

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 ...
3
votes
2answers
95 views

Localizations of $\mathbb{Z}/m\mathbb{Z}$

Let $A=\mathbb{Z}/m\mathbb{Z}$. Prove that for each multiplicative subset $\Sigma$ of $A$ there is an integer $n$ such that $\Sigma^{-1}A=\mathbb{Z}/n\mathbb{Z}$.
3
votes
1answer
142 views

Some questions about Fitting ideals

Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation $$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ...
3
votes
2answers
132 views

Subrings of rationals are noetherian

Let $R$ be a subring of $\mathbb{Q}$. Then $R$ is noetherian. Since $1 \in R$ it follows, that $\mathbb{Z} \subseteq R \subseteq \mathbb{Q}$. Consider an ideal $I \subseteq R$. Then $I':=I ...
3
votes
1answer
76 views

Is every prime ideal in $\Pi_{n=1}^{\infty}{k}$ maximal?

Suppose k is a algebraic closed field, is every prime ideal $\mathfrak{p}$ in the product ring $\Pi_{n=1}^{\infty}{k}$ maximal?
3
votes
3answers
192 views

Example of invertible maximal ideal that is not generated by one element

Could anyone give me an example of an invertible maximal ideal of some integral domain which is not generated by one element?
3
votes
1answer
240 views

When is the generic point of an integral noetherian scheme open (reference)?

Let $X$ be an integral noetherian scheme, let $\xi$ be its generic point. Then it is not so hard to show that $\{ \xi\}$ is open in $X$ if and only if $X$ is a finite set. In termes of algebra, it ...
3
votes
2answers
189 views

Counterexample to “if $f:A\to A$ induces the identity $\hat f:\operatorname{Spec}(A)\to\operatorname{Spec}(A)$, then $f=\operatorname{id}_A$” [closed]

Does anyone know any example that invalidates the following affirmation: If a morphism $f:A\to A$ induces the identity $\hat f:\operatorname{Spec} \left( A \right) \to \operatorname{Spec} \left( A ...
3
votes
1answer
137 views

Prime and Primary Ideals in Completion of a ring

Let $(R,\mathfrak m)$ be a local noetherian ring and $\widehat{R}$ its $\mathfrak m$-adic completion. If $\mathfrak q\in \operatorname{Spec}(\widehat{R})$ then can we find $\mathfrak p\in ...
3
votes
2answers
696 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 ...
3
votes
2answers
341 views

Factorization of ideals in $\mathbb{Z}[\sqrt{5}]$

Consider the ring $R=\mathbb{Z}[\sqrt{5}]$. Let $I$ be the following ideal of $R$: $$I:=(3,1+\sqrt{5})$$ My teacher said that the following equation holds: $$I^2=(3)I,$$ but I actually can't ...
3
votes
1answer
261 views

Nice proof for finite of degree one implies isomorphism?

Let $f: X \longrightarrow Y$ be a morphism of varieties over $\mathbb{C}$ and assume it is finite of degree 1, i.e. it is surjective and $$ [K(Y):K(X)] = 1 \quad \quad (*) $$ i.e. the function fields ...
3
votes
1answer
329 views

Coordinate ring of a cartesian product

I am considering the coordinate ring $k[X \times\mathbb{A}^n]$, where $X$ is an algebraic variety in $\mathbb{A}^n$. I want an isomorphism between this and the polynomial ring $k[X][y_1,\ldots, y_n]$. ...
3
votes
1answer
169 views

When are a certain types of rings UFDs

Recently, I learnt this nice theorem of Kaplansky: Let $R$ be an integral domain. Then $R$ is a UFD iff every nonzero prime ideal contains a nonzero prime principal ideal. Using this theorem, ...
3
votes
1answer
803 views

UFD implies noetherian?

It is easy to show that a PID must be noetherian. My question is: Does UFD imply noetherian? If not, is there an easy counterexample? I apologize if this turns out to be a simple question. ...
3
votes
2answers
161 views

$xy\in (x^2,y^2)$ if $R$ is a Dedekind domain

I would really like to see a simple proof for the following question, if possible. Let $R$ be a Dedekind domain. Then, $xy \in (x^2,y^2)R$ for any $x,y$ in $R$. Also, show that this fails in ...
3
votes
3answers
138 views

$\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ $\Rightarrow$ $R$ is a PID

Is the following true: If $R$ is a commutative unital ring with $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$, then $R$ is a PID. If yes, how can one prove it? Since $0$ is a prime ...
3
votes
2answers
340 views

How can I prove that every maximal ideal of $B= \mathbb{Z} [(1+\sqrt{5})/2] $ is a principal?

How can I prove that every maximal ideal of $B= \mathbb{Z} [(1+\sqrt{5})/2] $ is a principal? I know if I show that B has division with remainder, that means it is a Euclidean domain. It follows that ...
3
votes
0answers
176 views

Reflection of Exact Sequences

Consider the category of $R$-modules. I am trying to see how i can express a short exact sequence in terms of kernels and cockerels, and how this description can be used to prove that a conservative ...
3
votes
1answer
80 views

An ideal is homogenous iff it is invariant under a certain automorphism.

I'm working on the following. Let $R=R_0+R_1+ \cdots $ be a graded ring and $u$ a unit of $R_0$. Then the map $T_u$ defined by $T_u(x_0+x_1+ \cdots +x_n) = x_0+x_1u+ \cdots + x_n u^n$ is an ...
3
votes
1answer
448 views

Krull-Akizuki theorem without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. We use the definitions in my answers to this question. Can we prove the following theorem without ...
3
votes
1answer
236 views

The completion of a noetherian local ring is a complete local ring

We have defined the completion of a noetherian local ring $A$ to be $$\hat{A}=\left\{(a_1,a_2,\ldots)\in\prod_{i=1}^\infty A/\mathfrak{m}^i:a_j\equiv a_i\bmod{\mathfrak{m}^i} \,\,\forall ...
3
votes
0answers
129 views

Extension of the theorem of Jacobson

Let $A$ be a ring. Let $E$ be the set of polynomials $\{X^n-X \in \mathbb{Z}[X]|n \in \mathbb{N}^*-\{1\}\}$. By the theorem of Jacobson, we know that if for each $a\in A$ there is an element of $E$ ...
3
votes
1answer
235 views

Two questions about integral “splitting ring” extensions

We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains ...
3
votes
1answer
206 views

Poincaré series and short exact sequences

For an additive function $\lambda$ and an exact sequence of modules $0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$, we have $\lambda(M_2) = \lambda(M_1) + \lambda(M_3)$ by ...
3
votes
1answer
364 views

Integral closure of a local ring is the intersection of valuation rings lying above it

Let $L/K$ be a finite field extension. Let $\mathcal{O}$ be a valuation ring of $K$. Let $R$ be the integral closure of $\mathcal{O}$ in $L$. Why is $R$ the intersection of all valuation rings of $L$ ...