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

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10
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2answers
369 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
votes
1answer
269 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
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1answer
44 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
10
votes
2answers
148 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
249 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
votes
1answer
890 views

Intuition behind Hilbert's Nullstellensatz

maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof, however I cannot understand the concept in a more ...
10
votes
1answer
224 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
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2answers
1k views

When is a tensor product of two commutative rings noetherian?

In particular, I'm told if $k$ is commutative (ring), $R$ and $S$ are commutative $k$-algebras such that $R$ is noetherian, and $S$ is a finitely generated $k$-algebra, then the tensor product ...
10
votes
1answer
131 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
10
votes
1answer
125 views

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
10
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0answers
155 views

Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
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0answers
225 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
9
votes
3answers
1k views

Prime ideals in the ring of algebraic integers

Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$. It is known that $\mathcal{O}$ is a ...
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3answers
2k views

If a ring is Noetherian, then every subring is finitely generated?

Let $R$ be a commutative ring with $1$, and let $K$ be a field. We know that $R$ is Noetherian iff every ideal of $R$ is finitely generated as an ideal. Question 1: If $R$ is Noetherian, is every ...
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4answers
684 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 ...
9
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3answers
460 views

If $R$ is a commutative ring with identity, and $a, b\in R$ are divisible by each other, is it true that they must be associates?

Thank you very much! My problem is: If $R$ is a commutative ring with identity, and $a, b$ are its elements that are divisible by each other, is it true that they must be associates? Here, $a$ ...
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6answers
810 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 ...
9
votes
2answers
717 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 ...
9
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2answers
1k views

Is fibre product of varieties irreducible (integral)?

Let $k$ be an algebraically closed field and $X,Y$ varieties (i.e. integral, separated schemes of finite type over $k$). Is the fibre product $X \times_k Y$ necessary irreducible or integral? I ...
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votes
2answers
599 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
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2answers
436 views

A proof using Yoneda lemma

Martin Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of $$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong ...
9
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2answers
877 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
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2answers
299 views

$K[x_1, x_2,\dots ]$ is a UFD

I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field. If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization ...
9
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2answers
372 views

Is every prime element of a commutative ring “veryprime”?

Let $R$ denote a commutative ring. Define a function $$\| : R \times R \rightarrow \mathbb{N} \cup \{\infty\}$$ such that $a \| b$ is the number of times $a$ divides $b$ (and include $0$ in ...
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2answers
475 views

Injectivity of Homomorphism in Localization

Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}(Q)\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
9
votes
3answers
291 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
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2answers
2k views

Prerequisites for Atiyah Macdonald

I am currently doing a one semester course on groups and rings where we have learned about (so far): Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, ...
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votes
2answers
626 views

Hartshorne Exercise 1.1 (a)

(see bottom for apology) Let $Y$ be the plane curve $y = x^2$ (i.e., $Y$ is the zero set of the polynomial $f = y - x^2$). Show that $A(Y)$ is isomorphic to a polynomial ring in one variable over ...
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1answer
2k views

Definition of a finitely generated $k$ - algebra

In Miles Reid's Undergraduate Commutative Algebra he defines a ring $B$ to be finite as an $A$ - algebra if it is finite as an $A$ - module. Now what I don't understand is suppose we look at the ...
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3answers
689 views

Hartshorne Problem 1.2.14 on Segre Embedding

This is a problem in Hartshorne concerning showing that the image of $\Bbb{P}^n \times \Bbb{P}^m$ under the Segre embedding $\psi$ is actually irreducible. Now I have shown with some effort that ...
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votes
4answers
560 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
735 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
votes
2answers
154 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
9
votes
2answers
424 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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votes
2answers
167 views

Is every affine scheme the complement of the closed point $x$ of the spectrum of a local ring $A$?

Let $R$ be a commutative ring with identity element and let $\operatorname{Spec}(R)$ be the associated affine scheme. Does for each affine scheme $\operatorname{Spec}(R)$ exist a local ring $A$ ...
9
votes
2answers
260 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
249 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.
9
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2answers
129 views

$R/I$ is not Noetherian. Prove that $I$ is a prime ideal.

Let $R$ be a commutative ring with $1$ and let $I$ be an ideal of $R$, maximal with respect to the property that $R/I$ is not Noetherian. Prove that $I$ is a prime ideal. I need some hints to ...
9
votes
1answer
472 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
100 views

Direct proof of non-flatness

Consider $k$ a field and the rings $A=k[X^2,X^3]\subset B=k[X]$. How to prove that $B$ is not flat over $A$ by using only the definition of flatness that it maintains exact sequences after making ...
9
votes
2answers
348 views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
9
votes
1answer
255 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
votes
1answer
200 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
votes
1answer
576 views

Construct ideals in $\mathbb Z[x]$ with a given least number of generators

How do you construct, for each $n\geq 1$, an ideal in $\mathbb Z[x]$ of the form $(a_1,a_2,\dots,a_n)$ with $a_i\in \mathbb Z[x]$ such that it is impossible to have ...
9
votes
2answers
632 views

Rings whose spectrum is Hausdorff

Let $A$ be a commutative ring with $1$ and consider the Zariski topology on $\operatorname{Spec}(A)$. When will $\operatorname{Spec}(A)$ be a Hausdorff space? If $A$ has positive or infinite ...
9
votes
1answer
138 views

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
9
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2answers
451 views

Exactness of sequences of modules is a local property, isn't it?

It's well known, that passing to modules of fractions is exact, i.e. if $M'\xrightarrow{f} M\xrightarrow{g} M''$ is an exact sequence of $A$-modules ($A$ being a commutative ring with ...
9
votes
1answer
120 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
votes
1answer
487 views

Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
9
votes
2answers
2k 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 ...