Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
563
questions
7
votes
0
answers
486
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"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z}...
- 3,565
6
votes
1
answer
263
views
If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?
This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$,
...
- 2,783
6
votes
2
answers
451
views
Divisibility of the degree of an extension by the degree of its residual field
Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and $...
- 677
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votes
1
answer
2k
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$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?
Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...
- 2,783
5
votes
0
answers
829
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How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,063
5
votes
3
answers
3k
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Generalized Chinese Remainder Theorem
Let $U,V$ be submodules of a $R$-module $M$. Then the diagonal induces an isomorphism
$M/(U \cap V) \to M/U \times_{M/(U+V)} M/V.$
This is a (useful!) generalization of the Chinese Remainder Theorem ...
- 61.5k
0
votes
2
answers
355
views
Rank of a $ \mathbb{Z}_{p}[[T]] $ module
Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
- 1,211
0
votes
1
answer
425
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Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...
- 2,783
4
votes
2
answers
2k
views
Cohen-Macaulay sheaves which are not locally free
A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth_{\mathcal{O}_x}(\mathcal{F}_x) = \dim\mathcal{O}_x$ for any $x\in X$, where $\mathcal{O}_x$ is ...
- 2,444
113
votes
2
answers
12k
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How would you solve this tantalizing Halmos problem?
$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric ...
- 4,706
101
votes
31
answers
28k
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Errata for Atiyah–Macdonald
Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...
74
votes
1
answer
5k
views
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
- 61.5k
52
votes
9
answers
12k
views
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,246
47
votes
10
answers
15k
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Rings in which every non-unit is a zero divisor
Is there a special name for the class of (commutative) rings in which every non-unit is a zero divisor? The main example is $\mathbf{Z}/(n)$. Are there other natural or interesting examples?
- 2,942
43
votes
5
answers
3k
views
Explicit elements of $K((x))((y)) \setminus K((x,y))$
In an answer to the popular question on common false beliefs in mathematics
Examples of common false beliefs in mathematics
I mentioned that many people conflate the two different kinds of formal ...
- 64.8k
42
votes
4
answers
11k
views
Classification of finite commutative rings
Is there a classification of finite commutative rings available?
If not, what are the best structure theorem that are known at present?
All I know is a result that every finite commutative ring is a ...
- 2,186
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,743
36
votes
4
answers
2k
views
Rings for which no polynomial induces the zero function
For any commutative ring $R$ let $R[x]$ denote the ring of polynomials with coefficients in $R$. Any polynomial $p \in R[x]$ naturally induces a function $\hat{p} :R \rightarrow R$. In some cases, a ...
- 525
36
votes
4
answers
12k
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Flatness and local freeness
The following statement is well-known:
Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-...
- 2,837
34
votes
8
answers
4k
views
Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
33
votes
5
answers
3k
views
(Short) Exact sequences with no commutative diagram between them
This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...
- 2,195
33
votes
3
answers
6k
views
Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?
Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$,
decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective,
respectively, injective? --
And ...
- 19.5k
32
votes
6
answers
9k
views
What is the universal property of normalization?
What is the universal property of normalization? I'm looking for an answer something like
If X is a scheme and Y→X is its
normalization, then the morphism
Y→X has property P and any ...
- 23.7k
30
votes
2
answers
2k
views
When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
- 418
29
votes
5
answers
9k
views
Local complete intersections which are not complete intersections
The following definitions are standard:
An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X_1,\ldots, X_n]$. ...
- 303
29
votes
2
answers
5k
views
Examples of algebraic closures of finite index
So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
...
- 545
29
votes
4
answers
1k
views
Varieties where every algebra is free
I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
- 61.5k
28
votes
0
answers
839
views
The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...
- 33.9k
28
votes
3
answers
7k
views
Non finitely-generated subalgebra of a finitely-generated algebra
Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...
26
votes
3
answers
2k
views
Invariance of $\mathbb{Z}[x]$ under a self-equivalence of the category of commutative rings with 1
Let $\mbox{Rings}$ be the category of commutative rings with $1$.
Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ such that
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$
- 635
24
votes
4
answers
4k
views
Serre's theorem about regularity and homological dimension
One of the nicest results I know of is (Auslander-Buchsbaum-)Serre's theorem asserting that a (commutative!) local ring is regular iff it has finite global dimensional.
I'd like to ask a somewhat ...
- 46.8k
23
votes
2
answers
3k
views
Criteria for irreducibility of polynomial
If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best
- 669
23
votes
5
answers
5k
views
To prove the Nullstellensatz, how can the general case of an arbitrary algebraically closed field be reduced to the easily-proved case of an uncountable algebraically closed field?
In his answer to a question about simple proofs of the
Nullstellensatz
(Elementary / Interesting proofs of the Nullstellensatz),
Qiaochu Yuan referred to a really simple proof for the case of an
...
- 1,381
23
votes
0
answers
653
views
CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$
Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
- 17.1k
22
votes
8
answers
4k
views
Axiomatic definition of integers
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...
- 897
21
votes
4
answers
2k
views
Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
- 64.8k
21
votes
4
answers
5k
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The number of ideals in a ring
Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
- 1,861
19
votes
1
answer
808
views
Is the regularity of finitely generated rings decidable?
Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...
- 243
19
votes
4
answers
2k
views
What is the geometric object corresponding to a subalgebra in a polynomial ring
Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
- 1,961
19
votes
6
answers
2k
views
Nonfree projective module over a regular UFD?
What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...
- 64.8k
17
votes
4
answers
2k
views
Constructive proof that a kernel consists of nilpotent elements
I am interested in the following innocent looking statement:
Let $A \leftarrow R \rightarrow B$ be two homomorphisms of commutative rings. Assume that their kernels consist of nilpotent elements. ...
- 5,402
17
votes
3
answers
1k
views
Is a retract of a free object free?
I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
- 1,705
17
votes
3
answers
678
views
Existence of a ring with specified residue fields
Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I require ...
- 696
17
votes
6
answers
3k
views
What's an example of a transcendental power series?
Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does ...
- 3,254
16
votes
1
answer
2k
views
Questions about spectra of rings of continuous functions
I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
- 64.8k
15
votes
1
answer
1k
views
Is $R=k[x_1,\ldots]\to k[[x_1,\ldots]]$ a flat morphism? What about $R\to\hat{R}$?
Let $k$ be a field. For $R=k[x_1,\ldots]$ with countably infinite number of variables, [due to the discussion in the comments] we have to make the following distinction between $k[[x_1,\ldots]]$ and ...
- 2,956
15
votes
3
answers
4k
views
Elementary Luroth theorem proof?
Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
- 1,412
15
votes
1
answer
642
views
Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?
Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero.
When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...
- 2,783
14
votes
1
answer
2k
views
Some questions about the ring Z((x))
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
- 10k
14
votes
2
answers
1k
views
About integer polynomials which are sums of squares of rational polynomials...
I have the following question for which I haven't been able to find any reference or proof.
Suppose we know that a univariate polynomial $P(X)$ with integer coefficients is the sum of squares of two ...
