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

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2
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1answer
454 views

Localisation of an ideal

This should be quite easy, but somehow I can't find the proof. Let $P\neq Q$ be two maximal ideals in the commutative ring $R$. Then $P_Q=R_Q$. ($P_Q$ is the localisation of the R-module $P$ at $Q$ ...
0
votes
3answers
307 views

rank function on Spec (help with definition)

one definition of the line bundle over a ring is: a finitely generated projective A-module such that the rank function Spec A → N (positive integers) is constant with value 1. We call A itself the ...
1
vote
0answers
382 views

Picard group for dummies

a picard group is the set of isomorphism classes of invertible R-modules. I just read that phrase in the CRing project notes without further explanations: Here are my questions: 1-under which law (I ...
3
votes
2answers
416 views

Totally ordered abelian group

Let $\Gamma$ be a totally ordered abelian group (written additively), and let $K$ be a field. A valuation of $K$ with values in $\Gamma$ is a mapping $v:K^* \to \Gamma$ such that $1)$ ...
3
votes
1answer
152 views

Atiyah Ex5.29: Local ring of a valuation ring

Let $A$ be a valuation ring of a field $K$. Show that every subring of $K$ which contains $A$ is a local ring of $A$. This problem is already asked and answered at mathoverflow. But I can't ...
21
votes
1answer
506 views

functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define ...
8
votes
4answers
804 views

Non-Noetherian ring with a single prime ideal

My question: What are the most simple examples of a commutative ring R satisfying both of the following two properties: 1. R is not Noetherian. 2. R has exactly one prime ideal.
10
votes
1answer
1k views

(Ir)reducibility criteria for homogeneous polynomials

Suppose I have a homogeneous polynomial in at least 3 variables over some algebraically closed field (of characteristic 0, if need be). Question: How may I test — by hand — whether it is irreducible? ...
2
votes
1answer
548 views

What is a typical example of the tensor product of modules failing to be left exact?

I am looking for an example of an exact sequence of $R$-modules $$ 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 $$ and a $R$-module $N$, such that $$ 0 \rightarrow M' ...
6
votes
2answers
1k views

Ideal correspondence

I'm confusing the ideal correspondence theorem. Is the following right? Ideal correspondence: Let $f:A \to B$ be a ring homomorphism. Then there is a one-to-one order-preserving correspondence ...
0
votes
1answer
74 views

Integral extensions and number of generators

here's a doubt which arised from a previous question: Suppose $R$ is a ring and $S \subseteq R$ is a subring. Moreover suppose $R$ is integral over $S$ and $R$ is finitely generated as an $S$-module. ...
2
votes
2answers
200 views

Integral and semi-local ring

Let $R$ be a ring and let $S$ be a subring of R. If $R$ is a semi-local ring and $R$ is integral over $S$, why $S$ is semi-local as well?
3
votes
1answer
112 views

Elementary question about integral extensions

I'm reading page $59$ of Reid's "Undergraduate commutative algebra" book. In example (ii) it says, $k[x^{2}] \subset k[x]$ is an integral extension. How do we know this? I mean, in order to show ...
2
votes
1answer
619 views

Minimal generating sets of free modules, and endomorphisms of free modules

I know that it seems very loose as a title but I hope this post will be beneficial to all the forum members. One thing I like about free modules is that they help one define maps directly as we do in ...
1
vote
1answer
118 views

Ring of Invariant

Let $G \subset SL_2(\mathbb{C})$ be a finite subgroup acting linearly on $\mathbb{C}[X, Y]$. Then it is claimed that the ring of invariants $\mathbb{C}[X, Y]^G$ is always a hypersurface. I am not able ...
5
votes
1answer
155 views

Lang's “General Integrality Criterion”

Theorem 3.7 in the chapter on ring extension on page 352 of the latest edition of Lang's "Algebra" appears redundant in its phrasing to me. Specifically, if $g_s$ is a polynomial of total degree ...
5
votes
1answer
223 views

Equivalent condition for being a Jacobson ring

(Atiyah-Macdonald, Ex. 5.25) Let $A$ be a ring. Show that the following are equivalent: i) $A$ is a Jacobson ring; ii) Every finitely generated $A$-algebra $B$ which is a field is finite over ...
2
votes
1answer
124 views

Relation between different formulations of Nakayama's lemma

In Hulek's Elementary Algebraic Geometry, Nakayama's lemma is stated as follows: Let $A \neq 0$ be a finite $B$-algebra. Then for all proper ideals $m$ of $B$, we have $mA \neq A$. (Here, $A$ and $B$ ...
4
votes
3answers
947 views

Dedekind's theorem on the factorisation of rational primes

Let $K$ be an algebraic number field, and suppose its ring of integers is $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}_K$. Let $f \in \mathbb{Z}[X]$ be the minimal polynomial ...
6
votes
1answer
400 views

Integral Extension of a Jacobson Ring

Let $A \subseteq B$ be an integral extension. Show that if $A$ is a Jacobson ring, then $B$ is also a Jacobson ring. My trial: Let $q$ be a prime ideal in $B$, and let $p:=q^c=q \cap A$. Since ...
11
votes
1answer
929 views

Do localization and completion commute?

Let $A$ be a commutative ring and $\mathfrak{p}$ be a prime ideal of $A$. Under which assumptions for $A$ and $\mathfrak{p}$ does localization by $\mathfrak{p}$ and completion with respect to ...
1
vote
2answers
187 views

Extension of homomorphism

Let $A \subset B$(integral domain), $B$ is finitely generated over $A$. Let $y_1, \cdots, y_n \in B$ algebraically independent over $A$. Then homomorphism $f:A \to \Omega$(algebraically closed field) ...
32
votes
3answers
4k views

Reference request: introduction to commutative algebra

My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura ...
4
votes
2answers
199 views

Field of algebraic numbers over Q with p-adic value

Define $\overline{\mathbb{Q}} \subset \mathbb{C}$ to be the subset consisting of all complex numbers which are algebraic over $\mathbb{Q}$. We know that $\overline{\mathbb{Q}}$ is a countable field ...
3
votes
2answers
110 views

equality of modules

I'm reading a proof of Nakayama's theorem; it says at a certain step that: For $M$, a finitely generated module on a ring $R, N$ a submodule, and $I$ an ideal of the ring $R$: If $M = N + IM$, then ...
5
votes
1answer
424 views

fiber product of local artinian rings

Let $A,B,C$ be local artinian rings and $p : A \to C, q : B \to C$ local homomorphisms. Why is the fiber product $A \times_C B$ again a local artinian ring? It is easy to see that $P:=A \times_C B$ ...
2
votes
1answer
722 views

Nilradical that is a prime ideal

Let $R$ be a non-reduced commutative ring(not necessarily Noetherian) with unit. Let the nilradical $\mathcal{N}$ of $R$ be a prime ideal with the property that $\mathcal{N}^2=0$. Do we know about the ...
3
votes
2answers
284 views

Chinese remainder type theorem in Fulton's Algebraic Curves

The book "Algebraic Curves" by Fulton is available free for download on his website. On page 27, Fulton constructs an isomorphism which is used several times throughout the book. His construction is ...
4
votes
2answers
253 views

Localization of $\mathbb{C}[x,y]/(x^{3}-y^{3})$

Consider the ring $R=\mathbb{C}[x,y]/(x^{3}-y^{3})$ and let $S$ be the set of all non-zero divisors of $R$. How to find $S^{-1}A$? I guess the idea is to find a ring which is isomorphic to (or ...
0
votes
1answer
181 views

Exercise in Atiyah (localization)

Let me refer you to: http://www-users.math.umd.edu/~karpuk/chap3solns.pdf Page 2, ex. 4 Can you please explain the following step: $tb=f(s')b=s'b$ Why $f(s')=s'$ ? Thanks
1
vote
1answer
114 views

Direct sum of asssociated primes of a module

Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module. Denote the set of all associated primes of $M$ by $Ass(M)$. If $R= \oplus_{i=1}^{n} M_{i}$ where each $M_{i}$ is an ...
1
vote
1answer
813 views

finitely generated k-algebra and polynomial ring

Let $k$ be a field and let $A \neq 0$ be a finitely generated $k$-algebra, and $x_1, \cdots, x_n$ generate $A$ as a $k$-algebra. Is there any relationship(inclusion, homomorphism, etc.) between $A$ ...
6
votes
0answers
80 views

Weak Global Dimension and Global Dimension

Let $R$ be a commutative ring (not necessarily Noetherian) with unit. Is there an example such that weak global dimension of $R$ is finite but the global dimension of $R$ is infinite? Can we find such ...
4
votes
1answer
126 views

Subgroups of index 3 in $1+p\mathbb{Z}_p$

Let $p$ be a prime. I'm trying to compute the subgroups of index $3$ in $\mathbb{Q}_p^\times$ to enumerate some cyclic extensions using CFT. I've essentially reduced the problem down to finding the ...
11
votes
2answers
375 views

Minimal systems of generators for finitely generated algebras over commutative (graded) rings

Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that ...
2
votes
1answer
95 views

Converse of a result in a Noetherian ring involving height

Let $A$ be a Noetherian ring and $\mathfrak{p} \in Spec(A).$ If $x \in \mathfrak{p}$ then it is well-known that $ht_{A/(x)}(\mathfrak{p}/(x)) \leq ht_A(\mathfrak{p}) \leq ...
2
votes
1answer
118 views

Prime spectrum and the restriction

In Atiyah's commutative algebra, exercise3.21(ii) is the following: Let $f:A \to B$ be a ring homomorphism. Let $X=Spec(A)$ and $Y=Spec(B)$, and let $f^* : Y \to X$ be the mapping associated with ...
4
votes
2answers
371 views

Two ideals with equal radical in a noetherian ring

Let $A$ be a commutative noetherian ring with two ideals $I,J$ such that $\sqrt{I}=\sqrt{J}$. Does there always exist integers $p,q,r$ such that $$ I^p \subset J^q \subset I^r? $$
4
votes
2answers
677 views

elementary question about tensor product of modules

I'm a bit embarrassed to ask this, but I've gotten myself confused over what I think is a simple issue. Let $A$ be a local ring, $k$ its residue field, and $M,N$ finitely generated $A$-modules. An ...
4
votes
1answer
153 views

Generators of the coordinate ring for prime ideals

One of the ways I find more useful to check if a given ideal $I$ of $K[X_1,\ldots,X_n]$ is prime, is to look at the quotient ring $K[X_1,\ldots,X_n]/I$. If I'm able to show it is isomorphic to ...
1
vote
1answer
157 views

Flat ideals in a ring

If $A$ is an integral domain and $I$ is an ideal of $A$ generated by a regular sequence $f_1,\ldots,f_r$. Is $I$ flat (as an $A$-module)?
5
votes
3answers
363 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}}) ...
6
votes
1answer
279 views

Valuation ring of $k(x, y)$ of dimension $2$

My question is as follows: Given a field $k$, is it always possible to find a valuation ring of $k(x, y)$ of dimension $2$?
4
votes
1answer
94 views

minimal primes of a homogeneous ideal are homogeneous

I am trying to study the proof of this result. It appears as part 3 of the proposition on page 2 of the following document http://math.mit.edu/classes/18.721/projgeom6.pdf I understand everything but ...
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4answers
1k views

Localization of a prime ideal in $\mathbb{Z}/6\mathbb{Z}$

How can we compute the localization of the ring $\mathbb{Z}/6\mathbb{Z}$ at the prime ideal $2\mathbb{Z}/\mathbb{6Z}$? (or how do we see that this localization is an integral domain)?
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vote
1answer
141 views

Projective analog of an affine result

If $R$ is a commutative ring with identity, then $Spec(R)=\cup D(f_i)$ $i\in A$ (where $D(f)$ is the principal open set in $Spec(R)$ consisting of prime ideals of $R$ not containing $f$.) is ...
7
votes
1answer
2k views

Isomorphism in localization (tensor product)

Let $A$ be a commutative ring with $1$ and let $M,N$ be $A$-modules. Since there is a map $f: A \rightarrow S^{-1}A$, defined by $a \mapsto \frac{a}{1}$ then given any $S^{-1}A$-module we can view it ...
10
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1answer
1k views

Adjointness of Hom and Tensor

Could someone provide me a link to the proof of the adjointness of Hom and Tensor. I did an extensive google search but could not find anything self contained that presented the proof in full ...
5
votes
1answer
143 views

Necessity of the Noetherian condition to derive a result about associated prime ideals

Let $A$ be a Noetherian ring and $\mathfrak a$ be an ideal of $A$. Then it is well-known that the associated prime ideas of $\mathfrak a$ are those prime ideals that have the form $(\mathfrak a:x)$ ...
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2answers
2k views

Inverse Image of Maximal Ideals

Given a map of commutative rings with unit, it is often the case that the inverse image of a maximal ideal is not maximal. For example, consider the inclusion $\mathbb{Z} \subseteq \mathbb{Q}$. ...