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

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0
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1answer
79 views

Exterior product of Modules, problem wih tensor product

Let $X$ and $Y$ be schemes over a field $k$ and $p,q$ the projections of $X \times Y$ on $X$ and $Y$. Let $M$ and $N$ be modules on $X$ and $Y$. Then the exterior product $M \boxtimes N $ is defined ...
8
votes
1answer
372 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 ...
6
votes
4answers
274 views

$I$ is maximal $\implies I$ is prime

Been asked to show this is true with hints $R/I$ field $\Longleftrightarrow$ $I$ is maximal and $R/I$ integral domain $\Longleftrightarrow$ $I$ prime. Can you check this please, I have had a ten ...
0
votes
1answer
187 views

Jordan-Hölder factors of a finite length module

Suppose one has a local ring $(A,\mathfrak{m})$ and a finite length $A$-module $M$ with $\operatorname{supp}(M) = \{\mathfrak{m}\}$. Does $M$ have a composition series consisting only of ...
1
vote
1answer
77 views

Common regular sequence of ring and module

Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M\neq0$ a finite $A$-module. Suppose $$d=\min\{{\operatorname{depth}A,\operatorname{depth}M\}}\geq1.$$ Then does there always exists ...
4
votes
1answer
293 views

Proof for an integral domain involving subrings

Theorem: if $R,S$ are integral domains, $R\subset S$, and where $s_{1},...,s_{n}$ in $R_{S}$. Then there is a $m\in \mathbb{N}$ and $t_{1},..., t_{m}$ in $S$ (not all 0) so that $s_{i}N\subset N $ ...
2
votes
2answers
78 views

Product of a complete module and a finite module

Let $A$ be a commutative noetherian ring, $I$-adically complete (and separated) with respect to an ideal $I \subseteq A$. Let $M$ be a finite $A$-module, and let $N$ be an $I$-adically complete ...
6
votes
1answer
543 views

Prime ideals of the ring of rational functions

Let $A$ be a commutative ring with identity. If $f = a_0 + a_1 x + \cdots + a_n x^n \in A[x]$ is a polynomial, define $c(f) = A a_0 + A a_1 + \cdots + A a_n$ the ideal of $A$ generated by the ...
13
votes
1answer
377 views

History of Commutative Algebra

There are books of the history of Algebraic Geometry, there are also papers about it (All had done by J.Dieudonné). But I could not find any book or paper about the history of Commutative Algebra. ...
4
votes
1answer
448 views

generators of a prime ideal in a noetherian ring

Suppose $R$ is a Noetherian ring and $P$ is a prime ideal. If the number of generators of $PR_P$ as an ideal in $R_P$ is $n$, can we say anything about the number of generators of $P$ as an ideal of ...
4
votes
3answers
2k views

A proof that this set is an ideal of a commutative ring

This is a homework problem which I have worked hard on, but got stuck at the last step. Any assistance would be much appreciated. The problem is from Herstein's Abstract Algebra, 3rd ed., section 4.3, ...
7
votes
2answers
363 views

Extension and contraction of ideals in polynomial rings

Suppose $I$ is an ideal in a polynomial ring $R=k[x,y]$. Let $\overline{k}$ be the algebraic closure of $k$ and let $S=\overline{k} [x,y]$. Then is $IS\cap R=I$?
1
vote
2answers
86 views

How do we glue splittings together?

Let $M$ be a finitely-generated module over a Dedekind domain $R$. I need to show that $M = M_1 \oplus M_2$ where $M_1$ is torsion and $M_2$ is projective. It's clear we can do this locally: indeed, ...
2
votes
1answer
125 views

Projective syzygy vs. free syzygy

When referring to syzygies, some books refer to free resolution and some books refer to projective resolution. Are they equivalent in some sense? Is it true, for instance, that the $n$-th syzygy in a ...
6
votes
3answers
228 views

When is a local algebra reduced?

Let $k$ be a field and let $A$ be a local $k$-algebra which has finite dimension over $k$. Let $\mathfrak{m}$ be the maximal ideal of $A$ and let $k' = A / \mathfrak{m}$ be the residue field. For ...
1
vote
2answers
132 views

Calculation of radical ideal in $\mathbb Z_{36} $

Let $R$ be the ring $\mathbb Z_{36}$. How can I calculate $ \sqrt{\langle 0\rangle} , \sqrt{\langle 9\rangle} $?
0
votes
4answers
499 views

In a Noetherian ring, does every set of generators of an ideal have a finite subset of generators

In a Noetherian ring, every ideal is finitely generated. Suppose an ideal $I$ in a Noetherian ring $R$ is generated by a set of generators $S$. If $S$ is infinite, does it have a finite subset that ...
1
vote
1answer
77 views

Depth of finite projective modules over a nonlocal ring

Let $(A,\mathfrak{m})$ is a Noetherian local ring and $P\neq0$ is a finitely generated projective $A$-module. Then by Auslander-Buchbaum formula, $\operatorname{depth}P=\operatorname{depth} A$. But is ...
2
votes
1answer
137 views

A property of different in Dedekind domains

Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...
5
votes
3answers
258 views

What is an example of $\mathscr O_{Spec R}(U)\neq S^{-1}R$ for some $S$ consisting of the elements of $R$ not vanishing on $U$?

I've been meditating on the very basics of algebraic geometry, and in particular on how exactly $X=\operatorname{Spec} R$ relates to its structure sheaf $\mathscr O_X$. In these meditations, I've ...
1
vote
1answer
90 views

When does the fraction field of a ring have a non-trivial Galois extension

I have read this previous question on existence of a non-trivial Galois extension. I was wondering about the following situation. Suppose, $R$ is a domain that is not a field. When does the fraction ...
2
votes
1answer
183 views

If a local ring is $\mathfrak{m}$-adically complete is it also $I$-adically complete [duplicate]

Suppose $(R,\mathfrak{m})$ is a local ring and $I$ a proper ideal. If $R$ is $\mathfrak{m}$-adically complete is it also $I$-adically complete.
4
votes
1answer
223 views

Zero image of an element in the direct limit of modules

Let $\left(M_i,f_j^i\right)_{i,j \in I, i \le j}$ be a directed family of modules over some ring. Assume there is an index $k \in I$ such that there exists $x_k \in M_k$ whose image is zero in ...
11
votes
2answers
542 views

Hom and tensor with a flat module

Let $A$ be a commutative noetherian ring. Let $M, N$ be $A$-modules, and assume that $M$ is finite over $A$. Let $P$ be a flat $A$-module. Is it true that there is an isomorphism ...
2
votes
0answers
145 views

Integral closure under completion

Suppose $(R,\mathfrak{m})$ is a commutative local ring with identity and $I$ an ideal in $R$. If $I$ is integrally closed, does it follow that $I\hat{R}$ is integrally closed? If not, is this true ...
4
votes
2answers
120 views

Is there a good proof that all the polynomials in this family are irreducible?

Writing the few lines below in PARI/GP, one easily checks that the polynomials $$(X^3-4019680)-(a_2X^2+a_1X+a_0)$$ are all irreducible over $\mathbb Z$ when $a_0,a_1$ and $a_2$ are integers between ...
2
votes
0answers
224 views

A doubt about the proof of the fact that $\mathbb Z [(1+\sqrt{-19})/2]$ is a pid

I was reading that the proof of the fact that $R =\mathbb Z [(1+\sqrt{-19})/2]$ is a principal ideal domain from here It actually shows that $R$ is a Dedekind-Hasse domain, that is let $ \alpha , ...
2
votes
1answer
100 views

Free modules and the exactness of a sequence

When I read Thang Le's paper the coloured Jones polynomial and the A-polynomial of knots, it says in page 21 that: Since $R=\mathbb{C}[t^{\pm1}]$ is a PID, and $C$ is free over $R$. So if we tensor ...
3
votes
1answer
70 views

Does Ext commute with surjective scalar extensions?

Let $A$ be a ring, $I\subset A$ an ideal, $M$, $N$ $A$-modules such that $IM=0$ and $IN=0$. Then the modules extend to $A/I$-modules, and we have ...
1
vote
1answer
224 views

Finite flat algebras over Noetherian domains

Let $A$ be a Noetherian domain and $B$ a finite $A$-algebra containing $A$ as a subring. Suppose there is a number $n$ such that for every maximal ideal $\mathfrak{m}$ of $A$, $$\dim_{k(\mathfrak{m})} ...
1
vote
2answers
198 views

Scheme of dual numbers over a field

Let $k$ be a field and $D:=\operatorname{Spec}(k[t]/(t^2)$ the scheme of dual numbers over $k$. Then what is the fibre product $D \times_k D$ with itself over $k$? In other words, what is ...
2
votes
1answer
234 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
476 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 ...
11
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
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2answers
133 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
272 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
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 ...
1
vote
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 ...
8
votes
2answers
467 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
118 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
168 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
169 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
votes
2answers
697 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
100 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
150 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
405 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
157 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
148 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
68 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 ...