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

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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 ...
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1k views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
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5answers
961 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 ...
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4answers
224 views

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals.

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals. Help me some hints. I have no idea to start. ...
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844 views

Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
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An integral domain whose every prime ideal is principal is a PID

Does anyone has a simple proof of the following fact: An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
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2answers
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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 ...
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619 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.
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219 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 ...
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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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402 views

Is this ring Noetherian?

The subring of $\mathbb{C}[x,y]$ consisting of all polynomials $f(x,y)$ whose gradient vanishes at the point $x=y=0$. Is this ring Noetherian?
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Is a regular ring a domain

A regular local ring is a domain. Is a regular ring (a ring whose localization at every prime ideal is regular) also a domain? I am unable to find/construct a proof or a counterexample. Any help would ...
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176 views

How does this step in the proof of the structure theorem for f.g. modules over a Dedekind domain work?

I am trying to show that every finitely generated projective module $P$ over a Dedekind domain $D$ is a direct sum of (fractional) ideals. May's notes on Dedekind domains claim the result can be ...
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2answers
225 views

Exercise 3.15 [Atiyah/Macdonald]

I have a question regarding a claim in Atiyah, Macdonald. A is a commutative ring with $1$, $F$ is the free $A$-module $A^n$. Assume that $A$ is local with residue field $k = A/\mathfrak m$, and ...
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3answers
862 views

Commutative property of ring addition

I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring: Let $R\neq ...
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3answers
346 views

Commutative Algebra without the axiom of choice

It is well known that in a commutative ring with unit, every proper ideal is contained in a maximal ideal. The proof uses the axiom of choice. This fact, and others that are proved using essentially ...
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2answers
427 views

Inverse image of the sheaf associated to a module

In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows ...
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566 views

Are bimodules over a commutative ring always modules?

Let $R$ be a commutative ring. It is true that every module over $R$ is an $(R,R)$-bimodule. Is the converse true? In other words is it possible that there is an $R$-module where left multiplication ...
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1answer
563 views

Affine scheme $X$ with $\dim(X)=0$ but infinitely many points

As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ...
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2answers
146 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 $A$ ...
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Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?

Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
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126 views

Commutativity characterization?

Let $R$ be a ring (not necessarily unital) and for any $x\in R$ there is an integer $n \geq 2$ s.t. $x=x^2+\cdots+x^n.$ Does it imply that $R$ is commutative?
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Principal divisors

How can i calculate the principal divisor $(f)$ where $$f = \frac{(x^{3}-1)}{(x^{4}-1)}$$ with $f\in\mathbb{F}_2(x)$. I am recently reading about the subject, so i am looking for a simple solution ...
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503 views

A question about the tensor product of $\mathbb{Q}$

I'm reading this blog post about $\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q}$ and I have two questions about it: Is a simple tensor a tensor that cannot be written as a sum of tensors? On the first ...
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842 views

Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$?

If $\varphi:R^{m}\to R^{n}$ is an epimorphism of free modules over a commutative ring, does it follow that $m \geq n$? This is obviously true for vector spaces over a field, but how would one show ...
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2answers
212 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 ...
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1answer
109 views

How does Local Cohomology detect UFD?

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a ...
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671 views

what is the fraction field of $R[[x]]$, the power series over some ring?

I have a question similar to 74335. Let $R$ be an integral domain. Is there a nice description of the fraction field of the power series $R[[x]]$? I know that this field can be a proper subfield of ...
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2answers
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Suggestions for further topics in Commutative Algebra

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski's Main Theorem, which was the professor's original ...
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Projective Modules over the Ring of Trigonometric Functions

Let $ R = \mathbb{R}[ \cos x, \sin x] $ and consider the ideal $ \langle 1 - \cos x, \sin x\rangle $. Is this ideal a projective module over $R$ ?
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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 ...
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1answer
432 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 ...
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1answer
647 views

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

I read this proof that if $D$ is an integral domain and $D[X]$ is a principal ideal domain, then $D$ is a field. My question is if the requirements can be relaxed a bit, namely: Is it true that ...
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482 views

If A is noetherian, then Spec(A) is noetherian

Let A be a noetherian ring. How can I show that Spec(A) is noetherian? Also, is there a way to show this by showing directly that the closed sets in Spec(A) satisfy the descending chain condition? ...
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1answer
293 views

Is an ideal generated by multilinear polynomials of different degrees always radical?

Definition. A polynomial $f\in\Bbbk[x_0,\ldots,x_n]$ is called multilinear if $\deg_{x_i}(f)=1$ for each $0\le i \le n$. In other words, $f$ is linear in each variable. If $f$ is homogeneous of ...
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1answer
382 views

Associate prime ideals and exact sequences of $R$-modules

Let $R$ be a commutative unitary ring, consider the exact sequence of $R$ - modules $$ 0\rightarrow N\rightarrow M\rightarrow L\rightarrow0. $$ We know that $$\text{Ass}(M)\subseteq \text{Ass}(N)\cup ...
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1answer
355 views

What conditions guarantee that all maximal ideals have the same height?

It fails in general that all maximal ideals in a commutative ring with unity have the same height. It's easy to construct a counter-example when the ring is NOT an integral domain (consider the ...
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2answers
193 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
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302 views

The spectrum of a product of rings

Let $A$ be the product of a family $(A_i)_{i\in I}$ of commutative rings, and $c$ the canonical continuous map from the disjoint union $U$ of the spectra of the $A_i$ to the spectrum of $A$: $$ ...
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324 views

The Ring of Cauchy Sequences

Let $S$ be the ring of Cauchy sequences of $\mathbb{Q}$, i.e. $S=\{(a_n)\in\mathbb{Q}^{\mathbb{N}}|(a_n)\, \text{is a Cauchy rational sequence in the ordinary distance} \}$, $S$ is a subring of ...
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1answer
66 views

Disjoint standard open sets in Spec(R)

The following appeared as a homework problem last semester in Johan de Jong's algebraic geometry course at Columbia (http://www.math.columbia.edu/~dejong/schemes.html), described as "a bit of a ...
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416 views

An example of a commutative ring in which every primary ideal is prime

It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
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413 views

Geometric meaning of completion and localization

Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions. The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
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381 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 ...
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1answer
162 views

Recovering free modules from their projective limit

Let $\dotsc A_2 \to A_1 \to A_0$ be a sequence of surjective homomorphisms of commutative rings. Consider the projective limit $\varprojlim_i A_i$. If $S$ is an (infinite) set, then $\varprojlim_i ...
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Reduced rings and tensor products

I assume all rings are commutative with identity. Denote $A':=A/ \sqrt{0}$ for convenience. Question is simple: For a given ring $R$ and $R$-algebras $A,B$, does this isomorphism $(A\otimes_R ...
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Construction(s) of new integral domains from “old ones”

Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series ...
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158 views

What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
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199 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 ...
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Computing the “lying over”, “going up”, “going down” ideals.

For any commutative unital ring $R$ and an ideal $\mathfrak{a}$ of $R$, we shall denote $$\begin{align*} \mathrm{Spec}(R)&:=\{\text{prime ideals of }R\},\\ ...