All Questions
6,053 questions
32
votes
5
answers
9k
views
How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
- 151k
32
votes
2
answers
3k
views
Graded local rings versus local rings
A lot of times I see theorems stated for local rings, but usually they are also true for "graded local rings", i.e., graded rings with a unique homogeneous maximal ideal (like the polynomial ring). ...
- 10.7k
31
votes
8
answers
21k
views
Reference book for commutative algebra
I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be:
More comprehensive than Atiyah–Macdonald
More readable than Matsumura (maybe better ...
31
votes
2
answers
2k
views
Should Krull dimension be a cardinal?
A totally ordered finite set $\quad \mathcal P_0 \varsubsetneq \mathcal P_1\varsubsetneq \dots \mathcal \varsubsetneq \mathcal P_n \quad$ of prime ideals of a ring $A$ is said to be a chain of ...
- 47.5k
31
votes
1
answer
2k
views
Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?
This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
- 4,897
31
votes
0
answers
1k
views
On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
- 19.6k
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
30
votes
1
answer
14k
views
Rank of a module
What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...
- 2,857
30
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 ...
- 63.9k
30
votes
6
answers
8k
views
Algebraic stacks from scratch [closed]
I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
30
votes
2
answers
3k
views
Even XOR Odd Infinities?
Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
(http://en.wikipedia.org/wiki/Peano_axioms#First-...
- 687
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
5
answers
5k
views
Why does the (S2) property of a ring correspond to the Hartogs phenomenon?
Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this ...
- 5,052
29
votes
2
answers
5k
views
Regular, Gorenstein and Cohen-Macaulay
All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...
- 1,661
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
7k
views
Elementary proof of Nakayama's lemma?
Nakayama's lemma is as follows:
Let $A$ be a ring, and $\frak{a}$ an ideal such that $\frak{a}$ is contained in every maximal ideal. Let $M$ be a finitely generated $A$-module. Then if $\frak{a}$$M=M$...
- 1,329
29
votes
2
answers
2k
views
What are applications of commutativity theorems for rings?
Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the ...
- 50.6k
29
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 ...
29
votes
1
answer
1k
views
Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 41.9k
29
votes
0
answers
875
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.
...
- 35.2k
28
votes
5
answers
4k
views
Does Smith normal form imply PID?
Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...
user5810
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427
28
votes
1
answer
2k
views
SOS polynomials with integer coefficients
A well known theorem of Polya and Szego says that every non-negative univariate polynomial $p(x)$ can be expressed as the sum of exactly two squares: $p(x) = (f(x))^2 + (g(x))^2$ for some $f, g$. ...
- 1,703
28
votes
3
answers
3k
views
Why is "h" the notation for class numbers?
A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\...
- 50.6k
28
votes
5
answers
9k
views
Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"
The answer is yes: take $J=...
28
votes
4
answers
4k
views
What are traces?
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
- 953
28
votes
3
answers
3k
views
Equivalent definitions of invertible modules
Let $R$ be commutative unital ring, and $M$ an $R$-module. $M$ is called invertible (a.k.a. projective module of rank one), if it is finitely generated, and $M_{\mathfrak{p}} \cong R_{\mathfrak{p}}$ ...
- 523
28
votes
1
answer
1k
views
Algebraic dependency over $\mathbb{F}_{2}$
Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
- 583
28
votes
2
answers
3k
views
Maximal Ideals in Formal Laurent Series Rings?
Setup: Let $k$ be a field, let $n$ be a positive integer, and let $R := k[[x_1,\ldots,x_n]]$ denote the commutative ring of formal power series over $k$ in $x_1,\ldots,x_n$. We know that there is ...
- 283
28
votes
2
answers
2k
views
A sum involving roots of unity
Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...
- 2,183
28
votes
2
answers
863
views
$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
- 18.7k
28
votes
1
answer
1k
views
What are retracts of polynomial rings?
Is there a known example of a ring endomorphism $f: \mathbb{Z}[x_1, \ldots, x_n] \to \mathbb{Z}[x_1, \ldots, x_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring?
...
- 53.3k
27
votes
5
answers
3k
views
Class number measuring the failure of unique factorization
The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:
Is there a ...
- 4,593
27
votes
13
answers
4k
views
Homological algebra for commutative monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
- 27.5k
27
votes
5
answers
14k
views
Flat module and torsion-free module
All rings in this question are integral.
It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see here. ...
- 951
27
votes
2
answers
2k
views
Is every commutative ring a limit of noetherian rings?
Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do ...
- 4,416
27
votes
5
answers
3k
views
Algebraic description of compact smooth manifolds?
Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
- 6,546
27
votes
4
answers
3k
views
Nilradicals without Zorn's lemma
It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$.
Every proof I found (e.g. in the classical "Commutative Algebra" by ...
- 1,042
27
votes
1
answer
2k
views
What was commutative algebra before (modern) algebraic geometry?
Reading "H. Matsumura - Commutative Ring theory" I had the impression that the definitions were all made to mean something in algebraic geometry afterwards. I wonder what was commutative algebra ...
- 393
27
votes
3
answers
1k
views
Graded analogues of theorems in commutative algebra
Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word
commutative ...
27
votes
2
answers
1k
views
Limit of a series of singularities
The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
- 5,846
26
votes
5
answers
3k
views
Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated?
The question
Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is unusual for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than ...
- 27.7k
26
votes
4
answers
3k
views
Is a domain all of whose localizations are noetherian itself noetherian ?
Is a domain $D$, all of whose localizations $D_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ?
The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory:
Let ...
- 558
26
votes
2
answers
9k
views
Maximal ideals in the ring of continuous real-valued functions on ℝ
For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
- 6,734
26
votes
1
answer
4k
views
Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
- 363
26
votes
3
answers
2k
views
When does the converse to Schur's Lemma hold?
Let $R$ be a commutative ring, let $A$ be an $R$-algebra, and let $M$ be an $A$-module. If $M$ is simple, then End$_{A-mod}(M)$ is a division ring.
A common use is when $R$ is the complex numbers $\...
- 3,103
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
26
votes
3
answers
2k
views
Why the stable module category?
Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...
- 63.9k
26
votes
2
answers
4k
views
Why are injective modules more complicated than projective modules?
For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ...
- 2,040
26
votes
1
answer
2k
views
History of Koszul complex
This is a question about the history of commutative algebra. I'm curious why the Koszul complex from commutative algebra is called the Koszul complex? All of Koszul's early papers are about Lie ...
- 1,545