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

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2
votes
1answer
237 views

Isomorphism of tensor product

Let $k$ be a field and $A$ and $B$ be two commutative $k-$algebras. Furthermore, let $I$ be an ideal in $A$ and $N$ be a $A\otimes_kB$-module. Then is it true that $((A/I) \otimes_k B) ...
6
votes
1answer
537 views

Annihilator of quotient module M/IM

Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$? One inclusion is certainly true, but I ...
13
votes
2answers
1k views

Video lectures for Commutative Algebra

Are there any good video lectures for learning commutative algebra at level of Atiyah-Macdonald?
6
votes
2answers
137 views

Proving that $k[a,b,c,d,e,f]/(ab+cd+ef)$ and $k[x_1,x_2,x_3,x_4,x_5]$ are not isomorphic

How would you show that for a field $k$, the rings $k[a,b,c,d,e,f]/(ab+cd+ef)$ and $k[x_1,x_2,x_3,x_4,x_5]$ are not isomorphic, using methods that are algebraic? To be quite honest, I have no idea ...
1
vote
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 ...
3
votes
4answers
290 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
votes
1answer
316 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 ...
1
vote
1answer
64 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 ...
8
votes
2answers
493 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 ...
5
votes
2answers
120 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 $J\subset A$?
6
votes
0answers
177 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 ...
0
votes
2answers
185 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 $A_p$ ...
10
votes
2answers
782 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?
1
vote
1answer
103 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 ...
3
votes
1answer
155 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 $A_{\mathcal m}/{{\mathcal m}A_{\mathcal m}} \simeq A/{\mathcal m}$. ...
5
votes
2answers
445 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 ...
1
vote
3answers
161 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 ...
3
votes
1answer
164 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? ...
1
vote
0answers
85 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 ...
0
votes
1answer
68 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 ...
2
votes
1answer
101 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 ...
4
votes
2answers
190 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 ...
12
votes
1answer
1k 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 ...
3
votes
1answer
208 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) ...
6
votes
2answers
435 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
votes
2answers
255 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 ...
1
vote
1answer
297 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 $K[X,Y]$. ...
8
votes
3answers
413 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 ...
22
votes
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' ...
1
vote
2answers
112 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 ...
2
votes
2answers
138 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 ...
9
votes
1answer
581 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 ...
1
vote
2answers
182 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 ...
0
votes
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 ...
4
votes
2answers
279 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 ...
1
vote
1answer
77 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
votes
1answer
649 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
votes
3answers
289 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 $\dim R=0$ ...
2
votes
1answer
168 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 ...
0
votes
1answer
65 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 ...
12
votes
2answers
1k 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.
4
votes
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
votes
1answer
114 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
votes
1answer
206 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 ...
6
votes
2answers
171 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 ...
1
vote
1answer
69 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 ...
3
votes
2answers
359 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 ...
2
votes
2answers
1k 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?
6
votes
2answers
896 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 ...
1
vote
1answer
137 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 ...