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

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1
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3answers
166 views

The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
11
votes
2answers
2k views

Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
2
votes
1answer
109 views

Vanishing Ideal of a Linear Subspace

Let $F$ be an infinite field. Let $V$ be a subspace of $F^n$. Let $V^{\perp}$ be the set of all linear functionals $F^n \rightarrow F$ that vanish on $V$. Let $I(V)$ be the vanishing ideal of $V$, i.e....
17
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2answers
4k views

Why is the localization at a prime ideal a local ring?

I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is ...
7
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3answers
717 views

Is every Artinian module over an Artinian ring finitely generated?

I know that if $R$ is Artinian, then a f.g. $R$-module is Artinian. Is f.g. a necessary condition?
4
votes
2answers
436 views

Motivation behind the definition of Zariski tangent space

Intuitively I think of tangent space at a point as the set of all points lying in the tangent plane passing throug that point. Here is the definition of Zariski tangent space Let X be an ...
0
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1answer
52 views

Two different definitions of Derivation

Here are two definition of derivations Definition 1 Let $A \rightarrow B$ be a homomorphism of commutative algebras, and $M$ a $B$-module. We define the derivations $...
30
votes
3answers
1k views

Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
0
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2answers
37 views

Does the concept of localizing at an extension of a prime ideal make sense?

If $A,B$ are commutative rings with $1$, $p$ is a prime ideal in $A$ and $f:A\rightarrow B$ makes $B$ an $A$-algebra, I want to know if it is possible to define the localization $B_p$ of $B$ at the ...
22
votes
3answers
924 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
6
votes
4answers
355 views

Show that the ideal of all polynomials of degree at least 5 in $\mathbb Q[x]$ is not prime

Let $I$ be the subset of $\mathbb{Q}[x]$ that consists of all the polynomials whose first five terms are 0. I've proven that $I$ is an ideal (any polynomial multiplied by a polynomial in $I$ must ...
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1answer
355 views

$\mathbb{Z}[X]$ is noetherian

This follows from Hilbert's basis theorem, which is valid for polynomial rings over any noetherian ring. But is there a more elementary proof, knowing that $\mathbb{Z}$ is a PID (even a Euclidean ...
2
votes
3answers
320 views

Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$. I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
8
votes
3answers
1k views

Tensor product of a module with an ideal is isomorphic to their standard product

Let $A$ be a commutative ring and $M$ an $A$-module. Let $I$ be any ideal of $A$. We have an epimorphism $M \otimes_A I \rightarrow IM$. It seems to me that this is not in general an isomorphism. ...
5
votes
4answers
246 views

Ring of invariants of Klein Four group

Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial $...
0
votes
0answers
46 views

Show that $\mathcal{O}^+_K$ contains $\mathcal{O}_K$, and that the discriminant $\Delta(K)$ is the index $[\mathcal{O}^+_K : O_K]$. [duplicate]

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that $\mathcal{O}_K = \mathbb{Z}b_1+\cdots+\mathbb{...
0
votes
1answer
106 views

Is $\mathfrak I^n/ \mathfrak I^{n+1}$ defined when $\mathfrak I = (0)$

Just now I realized that I asked a stupid question.. Please ignore this question. While learning Associated graded modules, I defined associated graded module, which ended in this particular ...
3
votes
1answer
466 views

Use Nakayama's Lemma to show that $I$ is principal, generated by an idempotent. [duplicate]

Let $I$ be a finitely generated ideal in $R$, such that $I^2 = I$. Using the fact there exists $x\in R$ such that $e = 1 - x\in I$ and $xI = 0$, use Nakayama's Lemma to show that $I$ is principal, ...
1
vote
1answer
34 views

$pM \neq M$ $\stackrel{?}{\Rightarrow}$ $p M_p \neq M_p$

Let $A$ be a ring, $M$ an $A$-module and $p$ a prime ideal of $A$ such that $pM \neq M$. According to my intuition, it is not necessarily true that $p M_p \neq M_p$. Any counterexample?
5
votes
3answers
826 views

Does any integral domain contain an irreducible element?

Let $R$ be an integral domain which is not a field. Does $R$ necessarily have an irreducible element? I suspect the answer is no, but I couldn't find an example showing that...
4
votes
1answer
208 views

What has projectiveness to do with Cohen-Macaulay rings?

I read in Jacob Lurie's lecture notes that if $R=k[x_{1},\dots,x_{s}]/p$, and $R'=k[y_{1},\dots,y_{t}]$ injects into $R$ via Noether normalization such that $R$ is finite over $R'$, then $R$ is Cohen-...
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votes
1answer
181 views

show that M[X] is an R[X]-module.

Let $M$ be an $R$-module and let $M[X]$ be the set of polynomials in $X$ with coefficients in $M$. using the fact that $M[X]$ is an $R$-module with the obvious addition and scalar multiplication. ...
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votes
2answers
768 views

Some localization is not finitely-generated as an R-module

Let $R$ be an integral domain with field of fractions $K$, and let $f \in R$ be a non-zero non-unit. Prove that the subring $S=R[1/f]$ of $K$ is not finitely-generated as an $R$-module, using the fact ...
6
votes
1answer
286 views

Can infinitely many primes lie over a prime?

Let $R \subset S$ be an integral extension of domains and $\mathfrak p \subset R$ a prime ideal. Can it be the case that there are infinitely many distinct primes ${\cal P} \subset S$ such that ${\cal ...
5
votes
2answers
226 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 ...
8
votes
1answer
213 views

What does projective space classify?

Let $A$ be a ring and let $\mathbb{P}^n = \operatorname{Proj} \mathbb{Z} [x_0, \ldots, x_n]$. Question. What does $\mathbb{P}^n$ classify? In other words, is there some kind of algebraic structure (...
20
votes
1answer
637 views

Are finitely generated projective modules free over the total ring of fractions?

Let $Q(A)$ be the total ring of fractions of a commutative reduced non-noetherian ring $A$. Let $P$ be a finitely generated projective module over $Q(A)$ which is of constant rank (i.e. locally free ...
1
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0answers
49 views

Lorenzen embedding theorem for an $\ell$-group

The Lorenzen embedding theorem for an lattice-ordered group says that any lattice-ordered group can be embedded into a product of totally ordered group. What condition on lattice-ordered group makes ...
10
votes
2answers
754 views

Intersection of finitely generated ideals

Let $I$, $J$ be finitely generated ideals in a ring $A$ (commutative with identity). I know that the intersection need not be finitely generated: can somebody give me an example? Thanks.
2
votes
0answers
136 views

Minimal syzygies for polynomial ideals

Let $I$ be an ideal of $S=k[x_1,\dots,x_n]$. I am asked to find a minimal free graded resolution of $I$, by means of syzygy matrices. I suppose there has to be an algoritmic approach to it, provided ...
1
vote
1answer
84 views

Ideals / Direct sum decomposition

Let $u = (u_1 , \ldots , u_n ) \in \mathbb{A}^n$. Let $I$ be the ideal of $A = \mathbb{C}[x_1 , \ldots x_n ]$ generated by the elements $x_1 - u_1 , \ldots , x_n - u_n$. (i) Show that as a $\mathbb{C}...
3
votes
1answer
258 views

vector bundles on the affine line over a PID

Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial? For $R=k[X]$ this is true by the Theorem of Quillen-...
3
votes
1answer
486 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 $k[X][y_1,\ldots, y_n]$. ...
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votes
1answer
105 views

Noetherian modules

Question: Let $R$ be a Noetherian ring, and $M$ be an $R-$module, show that $M$ is Noetherian if and only if $M$ is finitely generated. This is a question on my homework, I'm really confused about ...
2
votes
1answer
68 views

practical condition for minimality in primary decomposition

Situation: $I$ is an ideal in a polynomial ring with a primary decomposition, not necessarily minimal (minimal=irredundant). I want to minimal-ize it. For any couple of primary ideals with the same ...
7
votes
2answers
293 views

Dimension of the total ring of fractions of a reduced ring.

Let $A$ be a commutative reduced ring (need not be noetherian). Let $S$ be the set of all non-zerodivisors of $A$. What is the Krull dimension of $S^{-1}A$ ?
4
votes
3answers
722 views

Noetherian Local Ring

I came across this old exam problem. If $R$ is a local Noetherian ring and $I$ is an ideal in $R$ such that $I^2=I$ then $I =0$. Any hint would be appreciated. I'm only familiar with what the ...
0
votes
1answer
208 views

Hypotheses of the Conormal Exact Sequence

On Wikipedia, in the description of the conormal exact sequence, it is described as arising from a closed immersion, which corresponds in the affine case to a surjection of algebras. However, in ...
5
votes
3answers
635 views

The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?

Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) \...
1
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3answers
658 views

Localization of Noetherian and Artinian Modules

Theorem: Let $R$ be a commutative ring with unity, and $S\subset R$ be a multiplicatively closed subset. If $M$ is a Noetherian (Artinian) $R$-module then $S^{-1}M$ is Noetherian (Artinian) $S^{-1}R$-...
5
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2answers
593 views

Companion Lecture Notes to Atiyah-MacDonald?

Is there a set of lecture notes that follow Atiyah-MacDonald and expand on the dense passages, point out typos and so forth?
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1answer
185 views

Prime ideal in a Dedekind domain

Let $\mathfrak p $ be a prime ideal in a Dedekind domain $O$ with field of fractions $K$. Define $$\mathfrak p^{-1}= \{x \in K: x\mathfrak p \subset O\}.$$ Let $\mathfrak a \subset \mathfrak p$ and $\...
4
votes
2answers
111 views

Zerodivisors and nilpotents in $A/I$

I'm studying primary decomposition in the case of polynomial rings with coefficients in a field. I have defined associate prime ideals of an ideal I as the radicals of the primary ideals appearing in ...
4
votes
1answer
674 views

Finite + surjective + projective implies flat?

Let $f: X \rightarrow Y$ be a morphism of irreducible projective varieties, that is both finite and surjective. Does this mean that it is flat? I have tried the following: By finiteness, the map is ...
1
vote
1answer
119 views

Generators for this ideal

I have a ring $R$ unitary and commutative with four elements and characteristic $2$. I have $$I=\{f \in R[X,Y]; f(t,t^2)=0\ \forall t \in R\}.$$ I have to find a finite number of generators for this ...
6
votes
1answer
180 views

A valuation ring

In Qing Liu, Algebraic Geometry and Arithmetic Curves, page 116, exemple 4.1.8, one has $\mathcal{O}_K$ a discrete valuation ring with uniformizing parameter $t$, $P\in\mathcal{O}_K[S]$ an Eisenstein ...
4
votes
4answers
3k views

A proof that shows surjective homomorphic image of prime ideal is prime

Let $A, B$ be commutative rings with $1_{A}, 1_{B}$. Suppose that $\mathfrak{p} \neq (1)$ is a prime ideal in $A$ with $\mathfrak{p} \supseteq \ker{\varphi}$ where $\varphi: A \rightarrow B$ is a ...
4
votes
4answers
154 views

Irreducible element of the ring.

Element $X_1 X_2 \cdots X_n - 1$ is irreducible in $K[X_{1},\ldots,X_{n}]$ for $n\ge 1$, where $K$ is a field. For $n=2,3$ it is easy to see that the element is irreducible but for higher value of $n$ ...
3
votes
2answers
462 views

Elements of the square of a prime ideal

Let $R$ be a commutative ring with unity, and let $\mathfrak{p} \subset R$ be a prime ideal. If $ab \in \mathfrak{p}^2$, does one of the following hold? $a \in \mathfrak{p}^2$; $b \in \mathfrak{p}^...
1
vote
2answers
1k views

Tensor products of fields

Let $K/F$ be a field extension. I am interested in the situation where there exists a field extension $L/F$ such that the ring $L \otimes_FK$ is not a field. If there exists $z\in K \setminus F$ ...