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

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4
votes
1answer
187 views

Can I assume $R$ is a local ring?

I should prove this statement: Let $R$ be a ring, $M$ be a $R$-module and $P$ a projective $R$-module of finite type. If $x=\sum_i m_i\otimes p_i$ is an element in $M\otimes_R P$ such that $\sum_i ...
5
votes
1answer
175 views

transitivity of finitely generated condition

Let $A \subseteq B \subseteq C$ be rings. I know that if $B$ is a finitely generated $A$-module and $C$ is a finitely generated $B$-module, then $C$ is a finitely generated $A$-module. (Proof is in ...
1
vote
1answer
358 views

Concept of Free module in Polynomial ring

I'm studying Atiyah's commutative algebra. I have a question with free modules and the kind of thing in polynomial ring. I wrote the following so it cannot be true facts. A free $A$-module is ...
1
vote
1answer
141 views

$Spec(R)$ Noetherian and going up theorem

Let $S \subseteq R$ be commutative rings with 1 and suppose $Spec(R)$ is a Noetherian topological space. How do we show that the number of each $T \in Spec(R)$ lying over $P \in Spec(S)$ is finite? I ...
2
votes
0answers
132 views

Twisted forms of a free module

Let $R$ be a ring an let $A$ and $B$ be two $R$-algebras. If $S$ is faithfully flat over $R$ then we say that $A$ is a $S$-twisted form of $B$ if $A\otimes_R S$ and $B\otimes_R S$ are isomorphic as ...
5
votes
2answers
771 views

I want a proof without using Nakayama's lemma

I am trying to understand Nakayama's lemma. It looks like some "fixed point theorem". Using Nakayama's lemma , I can easily solve the following question. I want another proof. Thanks. Let $A$ be a ...
5
votes
2answers
261 views

Going up theorem (basic question)

If $S \subset R$ are commutative rings with $1$ and $R$ is an integral extension of $S$ then they have the same dimension. Basically the proof uses the going up theorem. But I have a question about ...
4
votes
1answer
386 views

Finite ring extension and number of maximal ideals

I want to understand why the following is true: Let $S \subseteq R$ be commutative rings with $1$ and assume that $R$ is finitely generated as an $S$-module by at most $k$ elements. For every ...
7
votes
2answers
614 views

Does the following property of the direct limit of a direct system follow from the axioms for a direct limit?

Question: Does it follow from the axioms for a direct limit that if $\mu_i(x_i)=0$ then there exists $j \geq i$ such that $\mu_{ij}(x_i)=0$? Definitions and notation: (Atiyah MacDonald, chapter 2, ...
3
votes
2answers
164 views

Noetherian ring and the same prime divisors

Let $A$ be a Noetherian ring and let $x \in A$ be an element which is neither a unit nor a zerodivisor. Why the ideals $xA$ and $x^{n}A$ (where $n \in \mathbb{N}$) have the same prime divisors? i.e. ...
3
votes
2answers
566 views

Hilbert's Nullstellensatz: intersection over maximal ideals?

In Reid's commutative algebra, the author gives some exposition about the Nullstellensatz, $I(V(J))=\operatorname{rad}(J)$. But I can't understand it: The Nullstellensatz says that we can take the ...
4
votes
4answers
4k views

Can you construct a field with 4 elements?

Can you construct a field with 4 elements? can you help me think of any examples?
15
votes
4answers
751 views

Why is ideal more important than subring?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
4
votes
2answers
520 views

Help understand canonical isomorphism in localization (tensor products)

Let $M,N$ be $A$-modules and let $P$ be a prime ideal. Can someone please explain why the following isomorphism holds? $$(M \otimes_{A} N)_{P} \cong M_{P} \otimes_{A_{P}} N_{P}$$ Here's what I ...
4
votes
3answers
1k views

examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical example. Another ...
5
votes
2answers
346 views

Where is the Axiom of choice used?

In Reid's commutative algebra, there is a proof of equivalent conditions of Noetherian rings, especially (1) The set of ideals of $A$ has the a.c.c. $\Rightarrow$ (2) Every ideal in $A$ is finitely ...
2
votes
2answers
274 views

Prime ideals and Zorn's lemma

Let $A$ be a ring, $x$ a nonzero element of $A$ and consider the annihilator of $x$, i.e $Ann(x)$. Now let $S$ denote the collection of all prime ideals of $A$ containing $Ann(x)$. It can be shown ...
1
vote
1answer
47 views

$\alpha (M/N) = (\alpha M + N) / N$

I want to prove $\alpha (M/N) = (\alpha M + N) / N$, where $M$ is an $A$-module and $\alpha$ is an ideal of $A$. There will be many ways, for example, define a map $f:\alpha M + N \to \alpha (M/N)$ ...
5
votes
1answer
775 views

On prime ideals in a polynomial ring over a PID (from Reid's _Commutative Algebra_)

More general version of this is in Reid's undergraduate commutative algebra. Prime ideals of $B[Y]$ where $B$ is a PID are as follows: $0$, $(f)$ for irreducible $f \in B[Y]$, and maximal ideals $m$. ...
2
votes
1answer
116 views

Support of a module and ideals

Let $R$ be a Noetherian ring , $M$ a finitely generated $R$-module and let $J$ be an ideal such that $Supp(M) \subset V(J)$ where $V(J) = \{P \in Spec(R) : P \supseteq J\}$. How to show there exists ...
3
votes
2answers
707 views

Finding a primary decomposition

Let $k$ be a field, and $R=k[x,y]$. I'm supposed to find two different minimal primary decompositions of the ideal $(x^2y, y^2x)$. It's easy to see that one minimal primary decomposition is ...
2
votes
1answer
322 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
208 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
284 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
304 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
147 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
476 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
646 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
793 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? ...
1
vote
1answer
314 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' ...
5
votes
2answers
617 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
67 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
182 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
94 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
476 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
109 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
147 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 ...
4
votes
1answer
153 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
113 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
568 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 ...
5
votes
1answer
224 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 ...
8
votes
1answer
499 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 $\mathfrak{p}$ ...
1
vote
2answers
155 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) ...
27
votes
3answers
2k 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 ...
2
votes
2answers
159 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
100 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
321 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 $P:=A \times_C B$ ...
1
vote
1answer
616 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
254 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
241 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 ...