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

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10
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2answers
498 views

Prove that the kernel of a homomorphism is a principal ideal.

I have been having trouble with an exercise in my abstract algebra course. It is as follows: Let $f: \mathbb{C}[x,y] \rightarrow \mathbb{C}[t]$ be a homomorphism that is the identity on ...
10
votes
2answers
263 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 ...
10
votes
1answer
402 views

Atiyah-MacDonald help with exercise 5.10

This is an exercise from Atiyah-MacDonald, if someone can give an idea on how to prove that $a)\Rightarrow b)$: Let $f:A\rightarrow B$ a ring homomorphism. a) ...
10
votes
1answer
260 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
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2answers
1k views

Show $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain

I'm attempting to modify the proof the $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain to prove a similar result for $\mathbb{Z}[\sqrt{6}]$. The idea is to prove that $\mathbb{Q}[\sqrt{6}]$ is Euclidean ...
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2answers
346 views

Computing intersection multiplicity using Tor - explicit example

When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$: Let $k$ be an algebrically ...
10
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1answer
252 views

Minimal spectrum of a commutative ring

Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
10
votes
1answer
237 views

Do there exist polynomials $f,g$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ for $a,b,c$ given polynomials?

I want to prove something bigger than the problem in the title and I want to create a lemma that is useful for the solution of the problem. But I am unable to prove (or give a counterexample) the ...
10
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1answer
192 views

Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
10
votes
0answers
146 views

checking that an element of a module is zero, point-wise

Let $M$ be a module over a commutative ring $R$. Let $s \in M$ be an element such that for any $x \in \mathrm{Spec}\,R$, the image of $s$ in $M \otimes \kappa(x)$ is 0 (where $\kappa(x)$ is the ...
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votes
4answers
4k views

Example of modules that are projective but not free; torsion-free but not free

Free modules are projective, and projective modules are direct summand of free modules. Is there any example of projective modules that are not free? (I know this is not possible for modules of ...
9
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2answers
470 views

Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
9
votes
3answers
773 views

Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
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2answers
379 views

Projective module over a PID is free? [duplicate]

A common result is that finitely generated modules over a PID $R$ are projective iff they are free. Is the same true that an arbitrary projective module over a PID is free? I can't find this fact ...
9
votes
2answers
446 views

A non-nilpotent formal power series with nilpotent coefficients

Does anyone have an example of a formal power series $$p=a_0+a_1x+ a_2x^2 + \cdots \in R[[x]]$$ ($R$ is a commutative ring) all of whose coefficients $a_i$ are nilpotent in $R$ such that $p$ is not ...
9
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3answers
238 views

Zariski topology in the complex plane: an example

I want to find the closure under the zariski topology, of this set $ \left\{ {\left( {x,y} \right) \in {\Bbb C}^2 ;\left| x \right| + \left| y \right| = 1} \right\} $ I have no idea what I can do
9
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2answers
1k views

Methods to check if an ideal of a polynomial ring is prime or at least radical

I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...
9
votes
4answers
405 views

Equality of two notions of tensor products over a commutative ring

Let $R$ be a ring (not necessarily commutative), let $M$ be a right $R$-module and let $N$ be a left $R$-module. Then the tensor product $M \otimes_R N$ is an abelian group satisfying the universal ...
9
votes
2answers
306 views

Concrete examples of valuation rings of rank two.

Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise ...
9
votes
1answer
284 views

Primes in a Power series ring

Let $\mathbb Z$ be the ring of rational integers. Consider the power series ring $\mathbb Z[[x]]$. It is known that $\mathbb Z[[x]]$ is unique factorization domain. What are the primes in $\mathbb ...
9
votes
1answer
889 views

Inverse limit of modules and tensor product

Let $(M_n)_n$ be an inverse system of finitely generated modules over a commutative ring $A$ and $I\subset A$ an ideal. When is the canonical homomorphism $$\left(\varprojlim\nolimits_n ...
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votes
1answer
216 views

Geometrical interpretation of $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$, $X_i$ algebraic sets in $\mathbb{A}^n$

Edit: I should point out that I'm working over an algebraically closed field $k$. Let $X_1,X_2\subset\mathbb{A}^n$ be affine algebraic sets. Show that $I(X_1\cap X_2)=\sqrt{I(X_1)+I(X_2)}$. Show ...
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2answers
385 views

Why is a variety over a non-alg. closed field a hypersurface?

Exercise $3$ on page $8$ of Kunz's Introduction to Commutative Algebra and Algebraic Geometry is as follows: If the field $K$ is not algebraically closed, then any $K$-variety $V \subset A^n(K)$ can ...
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1answer
248 views

A question on faithfully flat extension

This question arose while reading page 116 of Red Book by Mumford. Let $B$ be a faithfully flat extension of $A$. Can I claim that $b \otimes 1 = 1 \otimes b$ in $B\otimes_A B$ if and only if ...
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1answer
442 views

Every maximal ideal is principal. Is $R$ principal?

Let $R$ be a commutative ring with 1. If every maximal ideal of $R$ is principal, is $R$ a principal ring?
9
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2answers
168 views

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions. I know that $0$ and $1$ are the solutions, but I can't proceed ...
9
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1answer
193 views

Noetherian ring whose ideals have arbitrarily large number of generators

Does a commutative ring satisfying the following two properties exist? All ideals are finitely generated; There are prime ideals with arbitrarily large (finite) minimal generating sets.
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2answers
511 views

Infinite product of fields

The main source of inspiration for this question is this excerpt Recall: An ultrafilter on the set X gives you a maximal ideal in the ring of all real-valued functions, and these are the only ...
9
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1answer
740 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? ...
9
votes
1answer
434 views

Why does the structure theorem for finitely generated modules over PIDs fail for arbitrary modules over a PID?

The proof that I know of the theorem goes like this: Any module $M$ is a quotient of a free module $F$ (over any ring). Any submodule $K$ of a free module $F$ over a PID $R$ is a free module, so in ...
9
votes
1answer
241 views

Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$

Let $A$ be a commutative ring with identity and, $\mathfrak{a}$ and $\mathfrak{b}$ ideals.I'm trying to find sufficient and necessary conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + ...
9
votes
1answer
223 views

If $I$ is a finitely generated ideal of $A[X]$, is $I\cap A$ necessarily finitely generated for a commutative unital ring $A$?

Let $A$ be a commutative ring with $1$ and $A[X]$ the ring of polynomials in one variable over $A$. Assume $I$ is a finitely generated ideal of $A[X]$. My question is Is $I\cap A$ necessarily ...
9
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2answers
1k views

Video lectures for Commutative Algebra

Are there any good video lectures for learning commutative algebra at level of Atiyah-Macdonald?
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1answer
177 views

Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!
9
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1answer
114 views

Curious about Hilbert-Zariski theorem involving homogeneous variety and set of zeroes.

I got myself in a confusing situation the other week while trying to read a bit of algebraic geometry. I'm hoping someone can pull me out. Suppose $k$ is a field, and $V$ a homogeneous variety with ...
9
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1answer
269 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
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2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
9
votes
1answer
125 views

diagonalizing a matrix over the $\ell$-adics

Let $M$ be a $2 \times 2$ matrix with coefficients in $\mathbb{Z}_{\ell}$ whose characteristical polynomial is $$ P(T) = T^2- (a+d) T + (ad-bc). $$ I've encountered the following assertion: If ...
9
votes
2answers
131 views

$AB=z \mathrm{Id}_n$ implies $z^m BA = z^{m+1} \mathrm{Id}_n$ for what $m$?

This question builds on a series of questions looking for elementary proofs that $AB=\mathrm{Id}$ implies $BA=\mathrm{Id}$, for $A$ and $B$ both $n \times n$ matrices over a commutative ring. First ...
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votes
1answer
155 views

Is there a characterization of integral domains in terms of the homomorphisms out of them?

In the $\mathbf{Set}$-concrete category of commutative rings, we can define that an object $A$ is a field iff for every homomorphism $f : A \rightarrow B$, precisely one of the following holds. $f$ ...
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votes
1answer
179 views

Lifting isomorphisms between derived categories

Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. I will denote by $D(A)$ and $D(B)$ the derived categories of unbounded complexes over $A$ and $B$. Suppose ...
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votes
1answer
391 views

Exercise 11.5 from Atiyah-MacDonald: Hilbert-Serre theorem and Grothendieck group

I don't understand Exercise 11.5 of Atiyah & MacDonald, which demands one elaborate upon or rephrase the Hilbert–Serre Theorem (11.1) in terms of the Grothendieck group $K(A_0)$. Here's ...
9
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1answer
403 views

How to tell if an element of a quotient ring is a zero divisor

I am looking at Hartshorne Example III.9.8.4., p260. He says that $a$ is not a zero divisor in $k[a,x,y,z]/I$, where $$ I = (a^2(x+1) -z^2, ax(x+1)-yz, xz-ay,y^2-x^2(x+1)). $$ Is there a good way to ...
9
votes
0answers
337 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
9
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0answers
289 views

Trivial intersection of algebraic sets?

The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
8
votes
4answers
422 views

A question on definition of field of fractions

Wikipedia defines the field of fractions of a domain as The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded. What does ...
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2answers
512 views

How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
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votes
6answers
342 views

Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal

In a commutative ring with an identity, every maximal ideal is a prime ideal. However, if a commutative ring does not have an identity, I'm not sure this is true. I would like to know the ...
8
votes
4answers
301 views

Subrings of fraction fields

Let $R$ be an integral domain and let $S$ be a ring with $R \le S \le \text{Frac}(R)$ (fraction field). Question: Is there a multiplicatively closed subset $U \subseteq R\setminus \{0\}$ such that ...
8
votes
5answers
925 views

Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian.

I am revising for my Rings and Modules exam and am stuck on the following two questions: $1.$ Let $M$ be a noetherian module and $ \ f : M \rightarrow M \ $ a surjective homomorphism. Show that $f ...