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

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17
votes
5answers
3k views

Showing the set of zero-divisors is a union of prime ideals

I'm working on an exercise from Atiyah and MacDonald's Commutative Algebra, and have hit a bump on Exercise 14 of Chapter 1. In a ring $A$, let $\Sigma$ be the set of all ideals in which every ...
1
vote
0answers
12 views

Trivial question about splitting fields

I'm having trouble with super basic ring theoretic concepts. Given some irreducible $f$ in $k[x]$, denote by $p$ the projection to the quotient $F$ by the ideal generated by $f$. I'm struggling to ...
1
vote
4answers
41 views

Even-odd multiplicative algebraic structure with idempotency?

What is the algebraic structure for the multiplications of even elements and odd elements? Please notice that $o*o=o$, $e*e=e$ (idempotency) and $o\not = e$. 1st structure is such that even times ...
4
votes
0answers
41 views

Vakil's definition of smoothness — what happens at non-closed points?

The following is definition 12.2.6 in Vakil's notes. A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by ...
0
votes
1answer
38 views

Manually computing ideal quotient $\langle x\rangle : \langle x y z \rangle$ in $k[x,y,z,o]$

Please explain this ideal quotient in $k[x,y,z,o]$: $$\langle x\rangle : \langle x y z \rangle=\{f\in k[x,y,z,o] : fg\in \langle x \rangle\quad\forall g\in \langle x y z \rangle \}$$ where ...
0
votes
1answer
69 views

Prove that these two fields are isomorphic.

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
1
vote
0answers
19 views

Prime and Maximal Ideals of $\mathbb{Z}[x]$ [duplicate]

Consider $R=\mathbb{Z}[x]$. Also let $p$ be a prime. Then we want to find all the prime and maximal ideals of $\mathbb{Z}[x]$. The prime ideals are $(0), (p), (x)$ and $(ap + bx)$. Then we see that ...
1
vote
1answer
46 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
0
votes
0answers
46 views

Groebner basis and prime ideals.

Let $I$ be an ideal in a polynomial ring $P = K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for ...
3
votes
0answers
34 views

Elementary divisors for chains of submodules

Given free modules $N \le M$ of finite rank over a PID $R$, it's well-known that there is a basis $\{x_1,\ldots,x_n\}$ of $M$ and there are $e_1,\ldots,e_n \in R$ such that $\{e_ix_i\mid e_i \neq ...
3
votes
2answers
77 views

Residue fields of schemes of finite type (over $\mathbb{Z}$)

Suppose $X$ a scheme of finite type over $\mathbb Z$. I want to prove that: (1) The residue fields of closed points of $X$ are finite; (2) For a given $q=p^n$ with $p$ prime, there is only a finite ...
15
votes
2answers
766 views

Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
2
votes
2answers
67 views

Homogeneous localization and regularity

Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I ...
4
votes
3answers
338 views

Represent localization as a direct limit

Let $A$ be a commutative ring with identity, $S\subset A$ a multiplicatively closed subset and $1\in S$. Does the equation $$S^{-1}A=\varinjlim_{s\in S}A_s$$ make sense? Here $A_s$ is the ...
0
votes
0answers
41 views

How to show for a f.g. graded ring $R$, $R^{(m)}$ is generated by degree $1$ for some $m$?

Let $$R=\oplus_{i\geq 0} R_i$$ be a graded ring, which is finitely generated as a $R_0$ algebra. Let $R^{(m)}$ be $\oplus_{i\geq 0} R_{mi}$. Then how to show that for some $m \in \mathbb{N}$, ...
0
votes
1answer
44 views

Poincare series and Hilbert polynomial of some graded modules [on hold]

Let $k$ be a field, and let $k[X, Y ]$ be the polynomial ring in two variables equipped with the usual grading such that $\deg(X) = \deg(Y ) = 1$. Consider the ideals $I = (X, Y^2)$ and $J = (X^2, ...
-1
votes
1answer
59 views

Example of submodule which has higher “rank” than the module

Given a ring $A$, and an $A$-module $M$. Let $x_1, . . . , x_n ∈ M$. Say $\operatorname{rank}(M) = n$ if for all $m ∈ M$ there exist unique elements $a_i ∈ A$, $i = 1, . . . , n,$ such that ...
7
votes
2answers
84 views

Are weakly étale ring homomorphisms of finite presentation étale?

Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat. Question. Are weakly étale ring ...
1
vote
1answer
15 views

Algorithm for computing the inverse limit of a finite inverse system

Let $k$ be a field (finite if you'd like), let $(I,\le)$ be a finite directed poset with $|I|=n$, and let $(A_i,f_{ij})_{i\le j\in I}$ be an inverse system of finitely generated, graded, commutative ...
2
votes
2answers
203 views

Example of an integral domain that is not integrally closed and having some localization which is also not integrally closed

Can anyone show an example of integral domain that is not integrally closed and also has one of its localization with respect to a maximal ideal not integrally closed?
1
vote
1answer
13 views

Kernel of $M\to M[U^{-1}]$ and primary decomposition of $(0)$

I am working on exercise 3.12 from Eisenbud's Commutative Algebra and I am having trouble parsing the question. Let $M$ be a finitely generated module over the Noetherian ring $R$. Given any ...
0
votes
0answers
28 views

If $f : M\otimes_A A/m \to N\otimes_A A/m$ is surjective , so is $f : M \to N$. [on hold]

Let $A$ be a local ring with maximal ideal $m$. Let $f : M \to N$ be a morphism of $A$-modules, where $N$ is finitely generated. Show that if the map $f : M\otimes_A A/m \to N\otimes_A A/m,\quad ...
1
vote
1answer
33 views

If I is an irreducible ideal, and P is a prime ideal, is (I+P)/P irreducible?

Let $A$ be a commutative ring with unit, and $P$ a prime ideal. My question is: If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample? ...
0
votes
1answer
73 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
4
votes
1answer
46 views

$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, ideal of $p$ in the local ring. What is $m_p/m_p^2$?

In Section 6.8 of Undergraduate Algebraic Geometry by Reid, the author proved the following Theorem: There is a natural isomorphism of vector spaces $(T_pV)^*\cong m_p/m_p^2$ where $^*$ denotes ...
6
votes
1answer
712 views

A question about a proof of Noetherian modules and exact sequences

I proved part (i) of the following: Proposition 6.3. Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then i) $M$ is Noetherian $\iff$ $M'$ ...
1
vote
0answers
35 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
0
votes
0answers
34 views

Every non-Noetherian module has a submodule maximal with respect to being not finitely generated. [duplicate]

Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated whenever $N<A\leq M$. The question is related to If $M$ isn't ...
2
votes
1answer
55 views

Is this ring extension flat?

Let $k$ be a field of characteristic zero and let $A$ be a finitely generated $k$-algebra. Let $B=A[x_1,\ldots,x_n]$ be the polynomial ring over $A$ and let $I \subseteq B$ be an ideal such that $B/I$ ...
1
vote
1answer
73 views

When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition ...
0
votes
0answers
94 views

Why is the map from $A^n$ to $M$ a surjective homomorphism?

How can one do the problem 1.3.11 b in Algebraic Geometry and Arithmetic Curves? I have read basics of commutative algebra but this one seems to be too difficult. Let $A$ be a commutative ring with ...
0
votes
0answers
17 views

Does Magma let you specify primary invariants?

I am cross-posting this question from scicomp.SE. The computer algebra system Magma can calculate primary invariants (i.e. a homogeneous system of parameters) in an invariant ring of a finite group ...
4
votes
2answers
293 views

Is the failure of MaxSpec to be functorial due to homomorphisms which take non-units to units?

I suspect this is easy, but that I'm missing something obvious: If a ring homomorphism $f:R\rightarrow S$ is such that a maximal ideal $\mathfrak m$ in $S$ does not have $f^{-1}(\mathfrak m)$ ...
13
votes
2answers
531 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
5
votes
1answer
353 views

Contraction of maximal ideals in polynomial rings over PIDs

Let $R$ be a principal ideal domain which is not a field, and let $M$ be a maximal ideal of the polynomial ring $R[X_1,\dots,X_n]$. If $n=1$ it is very easy to see that $M \cap R \neq 0$. Is this also ...
1
vote
1answer
48 views

Exercise from Kaplansky - Commutative Rings (1.2.3)

Exercise 3 in section 1-2: Let $R$ be an integral domain, $P$ a finitely generated non-zero prime ideal in $R$, and $I$ an ideal in $R$ properly containing $P$. Let $x$ be an element in the ...
0
votes
1answer
40 views

Ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary

Show that the ideal $(Y^2,X-YZ)$ is $(X,Y)$-primary in $K[X,Y,Z]$, where $K$ is a field. I got a hint that I need to use this property: Let $f:A\to B$ be a ring homomorphism. If $q$ is ...
1
vote
0answers
16 views

Algorithm for computing an inverse image

Let $k$ be a field (finite if you'd like), and let $f:A\to B$ be a map of graded, commutative $k$-algebras. Suppose further that $A$ is finitely generated and choose a presentation ...
3
votes
1answer
58 views

Blowing up an affine scheme at a regular point

I am reading Liu's Algebraic Geometry and Arithmetic Curves and get stuck at Lemma 8.1.2: Let $A$ be a Noetherian ring an define for an ideal $I \subset A$ the $A$-algebra ...
3
votes
2answers
130 views

If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated.

I'm answering this question: Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated but $A$ is finitely generated whenever ...
13
votes
2answers
310 views

“Graded free” is stronger than “graded and free”?

This topic suggested me the following question: If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a ...
2
votes
1answer
54 views

Equivalent condition for being a regular prime ideal

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\tp}{\tilde{\mathfrak{p}}}$ $\newcommand{\tA}{\tilde{A}}$ I have a question about Neukirch, Algebraic Number Theory, page 92. The problem is to show the ...
3
votes
1answer
45 views

Localizations of an Artinian ring are isomorphic to quotients.

If $R$ is an Artinian ring with $\{\mathfrak p_1,\ldots,\mathfrak p_n\}$ the set of prime and thus maximal ideals of $R$, is it true that $R_{\mathfrak p_i}$ (the localization at $\mathfrak p_i$) ...
3
votes
1answer
40 views

Localization $(R_{\mathfrak p})_{\mathfrak q}$ for an Artinian ring.

Let $R$ be an Artinian ring, and let $\mathfrak p$ and $\mathfrak q$ be distinct prime ideals of $R$. I have to prove that $(R_{\mathfrak p})_{\mathfrak q}=0$. What I have done is the following: ...
1
vote
1answer
53 views

$>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$

Let $R = k[x_1,\dots,x_t,x_{t+1},\dots,x_n]$ and $>$ a monomial ordering on $R$. Then $>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$ for all $1\leq i \leq t, t+1 \leq j ...
-1
votes
1answer
50 views

How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [closed]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
1
vote
1answer
32 views

Symbolic power of a prime ideal is primary

Let $A$ be a commutative ring, $S$ a multiplicatively closed subset of $A$. For any ideal $\mathfrak a$, let $S(\mathfrak a)$ denote the inverse image of $S^{−1}\mathfrak a$ under the localization map ...
0
votes
0answers
22 views

Intersection of affine subvarieties [closed]

If the ideals $I_i$ define irreducible subvarieties of an affine space, can the scheme defined by the ideal generated by finitely many of the $I_i$ contain a embedded component?
0
votes
1answer
106 views

The completness of ring and its power series ring

Let $R$ be a ring and $I$ an ideal of $R$. If $R$ is $I$-adically complete, why then $R[[x]]$ is $(IR[[x]]+(x)R[[x]])$-adically complete? (Matsumura, Commutative Ring Theory, Exercise 8.6.) Take ...
3
votes
2answers
83 views

Can a product of non-principal ideals be principal, in a local ring?

In a commutative ring $R$, we can define the ideal class monoid $\mathrm{ICl}(R)$ of $R$ as follows. The elements of $\mathrm{ICl}(R)$ are equivalence classes of invertible ideals (where an ideal $I$ ...