All Questions
5,985 questions
13
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answer
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Can an infinite commutative ring have a finite (but nonzero) number of non-nilpotent zero-divisors?
By a theorem of Ganesan, if a commutative ring not a domain has only finitely many zero-divisors, then the ring must be finite. (There are analogous results for non-commutative rings.)
There are ...
- 14.5k
2
votes
0
answers
348
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How much can we say about the number of nilpotents in a finite local commutative ring?
A commutative ring is local if it has a single maximal ideal. If the ring is finite, this implies that all elements are either units or nilpotents. Further, all finite local rings have prime power ...
- 14.5k
20
votes
1
answer
3k
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On a theorem of Jacobson
In a comment to an answer to a MO question, in which Bill Dubuque mentioned Jacobson's theorem stating that a ring in which $X^n=X$ is an identity is commutative (theorem which has shown up on MO ...
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4
votes
0
answers
179
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Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
- 30.5k
6
votes
2
answers
2k
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Cardinality of maximal linearly independent subset
M a finitely generated module over a commutative ring A. I can't think of an example of two maximal linearly independent subsets of M having different cardinality. I know that they all have the same ...
- 2,857
5
votes
2
answers
2k
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Is there a non-projective flat module over a local ring?
Is there a non-projective flat module over a local ring?
Here I assume the ring is commutative with unit.
- 12.3k
13
votes
0
answers
496
views
Are the supports of $Ext^i(M,N)$ eventually periodic?
Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules.
Question: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic ...
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9
votes
3
answers
1k
views
Vector spaces with natural bases
Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default ...
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1
vote
0
answers
534
views
Integral element in the quotient of a polynomial ring
Hello,
I'm writting a "report" (to learn) on algebraic geometry, and was looking to write a proof for the following statement :
Theorem : Let $K$ be an algebraically closed field and $C_1, C_2 \...
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11
votes
1
answer
1k
views
The associated prime ideals of $Ext^i_R(M,N)$
If R is a commutative noetherian ring, M and N are modules with M finite. It is well known in commutative algebra that $AssHom_R(M,N)=Supp(M)\cap Ass(N)$. But I want to know whether there is a ...
- 30.5k
2
votes
1
answer
1k
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For which rings does there exist an invertible Vandermonde matrix?
Suppose $R$ is a commutative ring, and $S \subset R^{n\times n}$ is an $R$-module. We are given $H_0,\dots,H_n \in R^{n\times n}$, and we know that for all $r \in R$,
$$H_0 + r H_1 + \dots r^n H_n \in ...
- 55.9k
5
votes
4
answers
388
views
Familiar equations in more general settings
What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...
- 5,227
4
votes
1
answer
914
views
Elements of trace zero in a field extension
Let $K=F_q$ and $F=F_{q^3}$, define the set A={$x \in F$ : $Tr_{F/K} (x)=0$}.
Is it true that for every $x \in A$ there are $y,z \in A$ such that $x=yz$?
- 55.9k
2
votes
0
answers
165
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Ideals weak equivalence and "finite" equivalence
Let $R$ be an order in a number field. Two $R$-ideals $I$ and $J$ are weak equivalent if there exist (necessarily invertible) ideals $X$ and $Y$ such that $I X=J$ and $J Y=I$.
This is equivalent to ...
0
votes
1
answer
630
views
Useless question on rank
What is the rank of $A^{n}$ if A is the zero ring? It's clearly not $n$ as many careless authors claim, since it's not even invariant. I don't think it's 0 either because it does have a linearly ...
- 33.8k
7
votes
2
answers
1k
views
Is the category of affine schemes (over a fixed field) Cartesian closed?
This is probably a trivial question, but I don't see the answer, and I haven't found it on Wikipedia, nLab, nor MathOverflow.
Let $\text{ComAlg}$ denote the category whose objects are commutative ...
CommunityBot
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1
vote
1
answer
219
views
Name for a module with only one associated prime
In EGA IV2, Def. 3.2.4, Grothendieck defines a quasicoherent sheaf over a locally Noetherian scheme to be "irredondant" if it has a unique associated point. Presumeably, a module over a Noetherian ...
- 20.8k
16
votes
1
answer
2k
views
Does ZF prove that all PIDs are UFDs?
Main Question:
Does ZF (no axiom of choice) prove that every Principal Ideal Domain is a Unique Factorization Domain?
The proofs I've seen all use dependent choice.
Minor Questions:
Does ZF + ...
- 856
11
votes
2
answers
470
views
Localizing at the primitive polynomials?
For any UFD $R$, the concept of a primitive polynomial (gcd of the coefficients is 1) makes sense in $R[x]$. The product of two primitive polynomials is primitive (Gauss's Lemma), and certainly 1 is a ...
- 4,736
5
votes
3
answers
4k
views
How to prove these two rings are not isomorphic
In fact, it is a simple problem. I just want to know whether there are some interesting proof.
$Z[x_1, x_2, ......, x_{n^2-1}]$ and $Z[y_{11}, ......, y_{1n}, y_{21}, ......, y_{nn}]/(det(y_{ij})-1))...
CommunityBot
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13
votes
3
answers
6k
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Ranks of free submodules of free modules [duplicate]
Possible Duplicate:
Atiyah-MacDonald, exercise 2.11
The following question came up during tea today.
Let $R$ be a commutative ring with an identity and let $M \subset R^n$ be a submodule. ...
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5
votes
0
answers
437
views
Primary decomposition for non-affine schemes
I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call ...
- 7,328
6
votes
1
answer
376
views
Checking locally whether a homomorphism is a localization
All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-...
- 27k
3
votes
2
answers
2k
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Finitely-generated algebra over Z
Let A be an artin ring which is also a finitely generated algebra over Z.
Show that $|A|<\infty$.
If A would have been a field then I know how to prove it. I know that A is a product of local ...
- 12.6k
9
votes
1
answer
2k
views
Symmetric polynomials theorem
Hello all, I would appreciate comments on the following question:
A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x_1,...,x_n]^{S_n} = k[s_1,...,s_n]$,...
- 1,304
12
votes
0
answers
529
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
6
votes
2
answers
610
views
What is the set of possible values of the degree of the sum of two algebraic numbers with fixed degrees?
This question is related to Degree of sum of algebraic numbers and algebraic numbers of degree 3 and 6, whose sum has degree 12.
In this last question I asked a very special case of the following ...
CommunityBot
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3
votes
3
answers
1k
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question about tensor of two fields
Let $k$ be a field, $L$, $H$ extension fields of $k$, and $G=L\otimes_k H$. I wonder why (I want to know the proof but I can't find) the prime ideal of $G$ must be maximal, and its properties:
a) if $...
- 14.5k
3
votes
1
answer
3k
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Isomorphism between direct sum of modules
Let $M$, $N$ be two modules over ring $A$. If $M\oplus M\cong N\oplus N$, can we conclude $M\cong N$? In the case that $M$, $N$ are completely decomposable (e.g. finite-length module by Krull-Schmidt ...
- 1,198
2
votes
0
answers
234
views
Flatness of module
$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{...
- 2,857
2
votes
0
answers
333
views
Localization of module
M an A-module, $S\subset A$ a multiplicative subset. Is it possible for $S^{-1}M$ to have an $S^{-1}A$-module structure satisfying $\frac{a}{1}\cdot\frac{m}{1}=\frac{am}{1}$ other than the "usuall" ...
- 63.1k
3
votes
2
answers
335
views
Gaps in Dimension Polynomials
There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a ...
- 21
1
vote
0
answers
268
views
Rational map defined over K leads to algebra question
Hello,
Concrete algebraic question : Let $K$ be a perfect field, $\bar{K}$ a fixed algebraic closure and let $f \in \bar{K}[x_1,\ldots,x_n]$. I was wondering when there exists another polynomial (non-...
- 11
13
votes
2
answers
5k
views
Primitive element theorem without building field extensions
Is there are nice way to prove the primitive element theorem without using field extensions?
The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over $F$...
- 156k
0
votes
1
answer
2k
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Dimension of tensor product of modules
$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ {B}(M\...
4
votes
2
answers
1k
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elementary classification of artinian rings
this may be too elementary for mathoverflow, but I'll give it a try.
rings are commutative here. it is well-known that every $0$-dimensional noetherian ring is artinian. the standard proof uses a ...
CommunityBot
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5
votes
2
answers
2k
views
Why is the fibered coproduct of affine schemes not affine?
I am confused about the following issue:
Let $X=SpecS$, $U_1=SpecR_1$, $U_2=SpecR_2$. and suppose we have maps $S \rightarrow R_1$, $S \rightarrow R_2$. Let $U_3=Spec (R_1 \otimes_S R_2)$. We have ...
- 39.3k
1
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0
answers
2k
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Formal power series ring & completion
I encountered the following passage in Matsumura's Commutative Ring Theory :
A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $M\subset B$ a maximal ideal, $\mathfrak{m}=M\cap A$. Then $(...
- 2,857
1
vote
2
answers
340
views
Infinite collection of elements of a number field with very similar annihilating polynomials
Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number
of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and
let $B$ be the subset of $R$ ...
CommunityBot
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13
votes
1
answer
990
views
Is -1 a sum of 2 squares in a certain field K?
Consider the field of fractions $K$
of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$,
where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables.
Clearly $-1$ is a ...
- 65.4k
2
votes
1
answer
429
views
0 dimensional Dedekind domain?
It seems that the ratio of those authors allowing a field to be a Dedekind domain to those who do not is almost 50 - 50. Why such a bewildering lack of consensus for such an elementary notion?
- 12.6k
16
votes
2
answers
2k
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Extra principal Cartier divisors on non-Noetherian rings? (answered: no!)
On the way to defining Cartier divisors on a scheme $X$, one sheafifies a presheaf base-presheaf of rings $\mathcal{K}'(U)=Frac(\mathcal{O}(U))$ on open affines $U$ to get a sheaf $\mathcal{K}$ of "...
CommunityBot
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9
votes
2
answers
2k
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Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
- 14.5k
2
votes
2
answers
349
views
Subrings of rational functions invariant under change of sign
Let $R$ be a ring generated by $k$ rational functions in the
variables $x_1,...,x_n$ over the real numbers.
Is there an algorithm that computes a set of rational functions
$f_1,...,f_l \in R$ which ...
- 12.3k
5
votes
1
answer
271
views
Feasibility of linear programs
It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?
- 201
5
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1
answer
443
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Does the free resolution of the cokernel of a generic matrix remain exact on a Zariski open set?
"Random" modules of the same size over a polynomial ring seem to always have the same Betti table. By a "random" module I mean the cokernel of a matrix whose entries are random forms of a fixed degree....
- 538
1
vote
2
answers
992
views
Hilbert Syzygy Theorem - Induction step
Does someone know in which books, lecture notes or ... I can find the induction step of the proof of Hilbert Syzygy Theorem? I'd only found the proof for R[x] (e.g. Weibel) and I haven't really an ...
CommunityBot
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6
votes
1
answer
1k
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Seeking examples or proof: injectivity of Cartan homomorphism for commutative rings?
This question is motivated by some issue raised by David Speyer in this question.
Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules ...
CommunityBot
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10
votes
5
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$K_{0}(R) =\mathbb{Z}$ but some f.g. projective not stably free?
This question is motived by this recent question.
$K_{0}(R)=\mathbb{Z}$ is often used as a euphemism for saying that every finitely generated projective module is stably free; however, there are some ...
CommunityBot
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4
votes
6
answers
665
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Number of A Subset of Monomials
I need to count the number of monomials of degree $n$ in $k$ variables, $x_1,\ldots ,x_k$, that contain at least one variable with a power of 1. The monomials need not include all the variables. ...
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