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

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1answer
65 views

A question about a ring ext. $k \subset R$ where $k$ is a field, $R$ is not a field, $Spec(R)$ consists of only closed points and is finite

It is a well know fact that if $k \subset R$ is an extension of rings such that $R$ is a finite dimensional vector space over $k$, then every point of $Spec(R)$ is closed (i.e., equivalently every ...
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4answers
269 views

Example of a reduced ring over a finite field satisfying some other conditions

What is an example for: An extension of rings $k \subset R$ where $k$ is a finite field, $R$ is a finite dimensional vector space over $k$, $R$ is reduced, and $R \neq k[r]$ for all $r \in R$. So, ...
4
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1answer
315 views

Computing with ideals: over $K$ or over $\mathbb{Q}\subseteq K$? does it matter?

I'm beginning to learn to use SINGULAR, the computer algebra system (CAS) for commutative algebra. NOTATION: If $K$ is a field of characteristic $0$, then $\mathbb{Q}\subseteq K$; otherwise ...
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1answer
63 views

Coming up with an example

What is an example of a finite type $\mathbb{Z}$-algebra $R$ which satisfies the following conditions: (1) There is no ring map from $R \rightarrow \mathbb{Q}$ (2) For every positive prime $p \in ...
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2answers
458 views

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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2answers
117 views

Is every ideal in $\hat{A}$ extended?

Let $A$ be a Noetherian ring, $I\subset A$ an ideal, $\hat{A}$ the $I$-adic completion. Is it true that every ideal of $\hat{A}$ is of the form $\hat{J}$ for some ideal $J\subset A$?
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0answers
165 views

When does base change preserves Homs

Let $A \to B$ be a ring homomorphism and $M,N$ be two $A$-modules. Consider the natural map $\alpha_{M,N} : \mathrm{Hom}_A(M,N) \otimes_A B \to \mathrm{Hom}_B(M \otimes_A B,N \otimes_A B)$ Consider ...
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2answers
165 views

Lifting back homomorphisms from localized modules

Refer to exercises 9, 10 of chapter 3 in Lang's algebra, page 167. In particular, let $A$ be a commutative ring, $p$ a prime ideal and $M, N$ $A$-modules. Then $M_p, N_p$ are the localized $A_p$ ...
8
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2answers
676 views

When is the integral closure of a local ring also a local ring?

Suppose $A$ is a normal local domain contained in a field $K$. Suppose $B$ is the integral closure of $A$ in $K$. Under what conditions on $A$ is $B$ local?
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1answer
97 views

Homomorphism in case of local ring

Let $A$ be a local ring and $\mathcal m$ the maximal ideal, considered as an $A$-module. Is then every $A$-module-homomorphism $\mathcal m \rightarrow A/\mathcal m$ equal to zero? Remark: I pose ...
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1answer
149 views

Isomorphism of First Ext groups

Let $A$ be a commutative ring with $1$ and $\mathcal m$ be a maximal ideal. One knows that then there is a canonical isomorphism $A_{\mathcal m}/{{\mathcal m}A_{\mathcal m}} \simeq A/{\mathcal m}$. ...
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2answers
396 views

Modules over $k[X,Y]$

Over a PID like $k[X]$, all (non-trivial) ideals are free and hence projective. But the ring $k[X,Y]$ is not a PID. Is it possible to describe all ideals of this particular ring which are projective ...
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3answers
156 views

Adjunction of a root to a UFD

Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My ...
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1answer
144 views

Do a matrix and its transpose have the same invariant factors over a PID?

I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors? ...
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0answers
63 views

What exactly is a “representation singularity”?

I've heard the term "representation singularity" in a few contexts about numerical instability of algorithms to find Gröbner bases, but I can't seem to find a precise definition for what it actually ...
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1answer
67 views

Is a special module homomorphism injective?

Let $I$ be an ideal in a ring $B$ with $I^2=0$. Furthermore one knows that one has a splitting $\alpha: B/I \rightarrow B$ of the natural projection. Let $M$ be a finitely generated module over ...
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1answer
98 views

Does the category of graded rings have limits?

Let $\mathfrak{C}$ be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it? I am particulary interested in power series rings over a field. Is there a ...
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2answers
186 views

Projective modules over $k[X,Y]/(X^3,Y^5)$

I'm searching for an example of a module, that is not projective for $k[X,Y]/(X^3,Y^5)$, but projective for the two subalgebras $k[X]/(X^3)$ and $k[Y]/(Y^5)$. (I don't think it is relevant, but in ...
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1answer
846 views

Does localisation commute with Hom for finitely-generated modules?

Question. Let $R$ be a ring, $\mathfrak{p}$ a prime, $M$ a finitely-generated $R$-module, and $N$ any $R$-module. Is the natural map $$\textrm{Hom}_R(M, N)_\mathfrak{p} \to ...
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1answer
183 views

every $I\!\trianglelefteq\!R$ is free $\Longleftrightarrow$ $R$ is a PID

In a discussion on MO, I found someone claiming the following: Proposition: For a commutative unital ring $R$, the following are equivalent: (i) every submodule of a free $R$-module is free; (ii) ...
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2answers
393 views

Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?

Let $R$ be a commutative ring, $\mathfrak{p}$ a prime ideal of $R$, $M$ a $R$-module, and $N$ a $R_\mathfrak{p}$-module. Why do we have this isomorphism? $$M \mathbin{\otimes_R} N \cong M_\mathfrak{p} ...
4
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2answers
225 views

Ideals generated by irreducible elements

Let $R$ be a UFD and $f_1,\dots,f_n$ be irreducible elements of $R$. Does it follows that the ideal $\langle f_1,\dots,f_n\rangle$ is a prime ideal?. If it's not true in general then is it true in ...
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1answer
263 views

Factoring polynomials in several variables over algebraically closed fields

This is a follow-up to Projective Spectrum of $K[X,Y]$ . I see why the given ideals are prime or even maximal, however, I have yet to prove that they in fact make up the entire spectrum of $K[X,Y]$. ...
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3answers
347 views

Exactness of a short sequence of quotient modules

Suppose R is a commutative ring with 1, I $\subset R$ is an ideal. We have R-Modules A, B and C with C being flat, as well as a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C ...
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4answers
2k views

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' ...
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2answers
106 views

Does localization satisfy this property?

Suppose $R$ is a ring and $S$ a multiplicative set in $R$. Then the localization $S^{-1}R$ satisfies the universal property that every element of $S$ maps to an invertible element in $S^{-1}R$ and if ...
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2answers
128 views

Restricting the normalization map to the preimage of the nonnormal locus

Let $\nu:\tilde{X}\rightarrow X$ be the normalization of an integral scheme $X$. Let $Y$ be the closed subset of $X$ where $\nu$ fails to be an isomorphism, endowed with its reduced subscheme ...
8
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1answer
524 views

Fraction field of the formal power series ring in finitely many variables

What is the fraction field of the formal power series ring over a field in finitely many variables $K[[X_1,\dots,X_n]]$? Is there a nice description for this field? When $n=1$, I know this is the ...
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2answers
172 views

Localisation contained in completion?

I'm working on an exercise in which I have to show that localising and completing are exact functors. More precisely I have a Dedekind domain $R$ and a prime ideal $\mathfrak{p}$ and I have to show ...
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1answer
81 views

On unique factorizations of ideals

Using standard notations, let $K$ be a number field and $S = \left\{p_{1}, ..., p_{n}\right\}$ a finite set of non-zero prime ideals of $K$. Let $a$ be a non-zero fractional ideal of $K$. Prove that ...
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2answers
274 views

Different version of Gauss's Lemma

Let $A$ be a domain with field of fractions $K$. Let $f, g \in A[X]$ with $g$ monic. Show that if $f/g \in K[X]$ then $f/g \in A[X]$. So I tried the direct approach by just assuming $f/g$ has a ...
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1answer
76 views

$(a_1,\ldots,a_n)\!=\!(a)\;\Leftrightarrow\;a\!=\!\gcd(a_1,\ldots,a_n)$?

Could you please help me finish the proof below. The only problem is the $(\Leftarrow)$ part of a). Proposition???: In any domain: a) ...
2
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1answer
503 views

Left exactness of inverse limit

Is the left exactness of inverse limit (in the category of modules over a ring) a general property regardless of the indexing set? (Let's assume it is still directed.) The only proof I can find ...
3
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3answers
274 views

A Noetherian Ring with Discrete Spectrum is Artinian

I'm trying to solve an exercise. I should prove that if $R$ is a notherian ring and $\operatorname{Spec}(R)$ is discrete then $R$ is artinian. I think it is enough to show that $\dim R=0$ ...
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1answer
138 views

Quotient Rings and Integral Extensions

Suppose $S$ is an integral extension of $R$ and $I$ an ideal in $S$. Why is $S/I$ an integral ring extension of $R/(R \cap I)$? To this question, Dummit and Foote says: Reducing the monic ...
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1answer
64 views

intersections of powers of primes lying over a prime in a Galois extension

Suppose $A$ is a Dedekind domain with fraction field $K$ and $L/K$ is Galois, let $B$ be the integral closure of $A$ in $L$. Let $P$ be a prime ideal in $A$ and let $P_1,...,P_n$ be prime ideals ...
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2answers
913 views

A non-noetherian ring with all localizations noetherian

If for a ring $A$ every localization $A_\mathfrak{p}$ by a prime $\mathfrak{p}\subseteq A$ is noetherian, is it true that $A$ is noetherian? I believe not but I can't find a good counterexample.
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1answer
84 views

algebra homomorphism $k^S \to k$

Let $k$ be a field and $S$ be an infinite set. Assume $|S| \leq |k|$. Why is then every $k$-algebra homomorphism $k^S \to k$ equal to a projection $\mathrm{pr}_s$ for some $s \in S$? I don't know how ...
3
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1answer
107 views

Containment of primary ideals

Suppose, $R$ is a noetherian ring. Let $P$ be a prime ideal in $R$. Let $Q$ be a $P$-primary ideal that contains $P^n$. Then does $Q$ contain $P^{(n)}$ which is the $n$th symbolic power of $P$ and is ...
3
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1answer
176 views

Existence of inverse limit in an arbitrary category

According to Wikipedia article http://en.wikipedia.org/wiki/Inverse_limit "Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category." But when constructing the ...
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2answers
160 views

Associated primes of a sum of modules

Let $M$ be a module with $M_1$ and $M_2$ submodules such that their sum (not necessarily a direct sum) is $M$. Is it true in full generality that $\text{Ass}(M) = \text{Ass}(M_1) \cup ...
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1answer
64 views

Radical of an ideal after adjoining roots

Let $A$ be a Noetherian domain containing an algebraically closed field $k$. Let $x_1,\ldots,x_r\in A$ be irreducible elements generating a radical ideal $I=(x_1,\ldots,x_r)$. Set ...
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2answers
317 views

Projective Spectrum of $K[X,Y]$

Let's assume that $K$ is algebraically closed. I'm having some difficulties figuring out what $\text{proj}\;K[X,Y]$ is, where $K[X,Y]$ is interpreted as a graded ring. Any hints? So far I have only ...
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2answers
854 views

Ideal generated by a irreducible element

Is the ideal generated by an irreducible element always a prime ideal in a ring? If so why?
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2answers
849 views

A Question about Dedekind Domains

In this question I will use the following definition of a Dedekind domain: An integral domain $A$ is a Dedekind Domain if: 1) $A$ is a Noetherian Ring. 2) $A$ is integrally closed. 3) Every ...
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1answer
129 views

About injective hull of residue field

Let $(A,\mathfrak{m})$ be a noetherian local ring, and $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. I'm pretty sure that $E(A/\mathfrak{m})$ doesn't automatically extend to an ...
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1answer
622 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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2answers
378 views

Motivation behind the definition of Localization of rings

What is the motivation behind definition of localization of rings? From where does the term "localization" come from? Why is the equivalence relation between the ordered pairs ...
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1answer
240 views

epimorphism in the category of commutative rings

Let $\phi:A\to B$ be an epimorphism in the category of commutative rings, we can find that the induced continuous map $\phi^*$ from Spec$B$ to Spec$A$ is injective as a map between sets, I want ...
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1answer
150 views

Particular case of Nakayama's lemma

I'm trying to prove the following particular case of Nakayama's lemma. Let $R$ be a commutative ring and $a\in R$ be nilpotent (let's suppose $a^{k-1}\not=0$, $a^k=0$). Then $aM=M \Rightarrow ...