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

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35
votes
0answers
949 views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
5
votes
1answer
352 views

Non-Noetherian rings with an ideal not containing a product of prime ideals

It is well-known that in every commutative Noetherian ring every ideal contains a product of prime ideals. Are there examples of non-Noetherian rings with an ideal that does not contain any prime ...
8
votes
1answer
467 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 $A$: $$ ...
4
votes
1answer
128 views

Two isomorphisms of (algebraic) inverse limits

I am having trouble seeing why the following two isomorphisms should hold for a Noetherian ring $A$ and ideals $I$,$J$ of $A$: $$\varprojlim A/(I+J)^n \cong \varprojlim A/I^n+J^n$$ $$\varprojlim_m ...
2
votes
0answers
236 views

Depth of monomial ideal

Let $S=K[X_1,...,X_n]$, $K$ a field, and $I \subset S$ a monomial ideal generated by squarefree monomials of degree $\geq d$. Then show that depth$_S I \geq d$.
0
votes
1answer
49 views

Show that $A(V(f))/(X,Y)^n A(V(f))$is a local ring where $A(V(f))$ is the coordinate ring of an irreducible polynomial $f$ in $K[X,Y]$

Let $K$ be an algebraic closed field, $f$ is an irreducible polynomial in $K[X,Y]$,and $f(0,0)=0$. Denote $A(V(f))$ as the coordinate ring $K[X,Y]/(f)$. Now I don't konw how to show that ...
3
votes
0answers
309 views

What does a minimal generator of an ideal mean?

For example in lemma 2.1 here Another example is here the phrase "a minimal generator" is used. I don't understand what this means in the absence of a specified set of generators. Can anyone explain ...
10
votes
2answers
944 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 ...
1
vote
0answers
67 views

Can the completion of a non-domain be a domain

Suppose, $(R,m)$ is a Noetherian ring that is not a domain. Can $\hat{R}_m$ be a domain? I think this cannot be the case for if $a,b\in R$ s.t. $a,b\neq 0$ and $ab=0$. Then, this $R$ is a Noetherian ...
5
votes
2answers
387 views

Exercise 9.5.21 in Grillet (Jacobson radical, Nakayama's Lemma)

I've managed to prove the following exercise: (21.) Let $\mathfrak{m}$ be a maximal ideal of a commutative ring $R$. Prove the following: if $A$ is a finitely generated $R$-module, and $x_1, x_2, ...
1
vote
0answers
47 views

Computability of “isomorphism existence” between special cubic number fields

Let $a$ be a rational number such that the polynomial $P_a=X^3-X-a$ is irreducible, let $\alpha_{a}$ denote a root of $P_a$ and let ${\mathbb K}_a={\mathbb Q}(\alpha_{a})$. Similarly, let $b$ be a ...
0
votes
1answer
81 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
491 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
296 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
204 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
81 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
304 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
1answer
82 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 ...
8
votes
1answer
703 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 ...
16
votes
1answer
544 views

History of Commutative Algebra

There are books on 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
600 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, ...
10
votes
2answers
583 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
89 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
183 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
243 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
134 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
742 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
79 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
148 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
287 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
95 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
218 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.
5
votes
1answer
305 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 ...
14
votes
2answers
886 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
161 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
121 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
243 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
105 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
89 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
245 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
250 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
247 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
695 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 ...
14
votes
2answers
2k views

Video lectures for Commutative Algebra

Are there any good video lectures for learning commutative algebra at level of Atiyah-Macdonald?
7
votes
2answers
142 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
66 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
335 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
323 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
65 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 ...