All Questions
Tagged with rings-and-algebras or ra.rings-and-algebras
3,500 questions
7
votes
4
answers
2k
views
Are all parametrizations via polynomials algebraic varieties?
Suppose that we have a parametrization via polynomials as follows:
$$t\longrightarrow (f_1(t),\ldots,f_n(t)),$$
where $t$ is a vector in $\mathbb{C}^r$ and $f_i$ are polynomials of arbitrary degree.
...
- 2,215
8
votes
2
answers
1k
views
What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?
Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...
- 7,197
7
votes
3
answers
744
views
Looking for applications of a nice result in linear algebra
Hello everybody
There is a nice classical result in linear algebra: if $A, B$ are two matrices in $M_n(k),$ where $k$ is a field, and $B$ commutes with every element of $M_n(k)$ which commutes with $...
- 73
3
votes
2
answers
459
views
Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups
In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:
&...
- 2,858
1
vote
0
answers
2k
views
Generators of ideals in polynomial rings over commutative rings.
This is my first question; I hope it worthy of this awesome forum and its members.
Let $R$ be a commutative ring, perhaps with unit, perhaps not. As usual let $R[x]$
denote the ring of polynomials ...
- 391
5
votes
1
answer
410
views
Behavior of the projective dimension of modules in a continuous chain of extensions
Let $R$ be an arbitrary ring. Let $D$ be the class of $R$-modules of projective dimension less than or equal to a natural number $n$. If $L$ is the direct union of a continuous chain of submodules ${...
- 111
3
votes
0
answers
614
views
nilpotent matrices over polynomial rings
I am looking for an analogue of the Jordan normal form for nilpotent matrices over the
polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action ...
- 6,214
25
votes
6
answers
7k
views
prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,053
5
votes
2
answers
1k
views
Is there a notion of congruence relation for essentially algebraic structures?
In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations.
A congruence ...
- 12.3k
8
votes
2
answers
1k
views
Criterion for an abelian group to have a commutative endomorphism ring
Given an abelian group $G$, one can form the endomorphism ring $\mbox{End}(G)$ by letting $\alpha+\beta=\alpha(x)+\beta(x)$, and $\alpha\beta=\alpha(\beta(x))$, where $\alpha$ and $\beta$ are ...
- 5,739
6
votes
1
answer
317
views
sheaves of modules on an $\ell$-space
Let $X$ be a Hausdorff, locally compact, and totally disconnected topological space, which I call an $\ell$-space, and write $A = C^{\infty}_C(X)$ for the algebra of locally constant complex-valued ...
- 3,605
4
votes
2
answers
413
views
Universal functors according to Cohn.
In section III.1 of P.M. Cohn's Universal Algebra a notion of universal functor ${\cal L} \rightarrow {\cal K}$ is defined for a representation of one category in another given by a (covariant) ...
- 879
3
votes
2
answers
536
views
An example of a Z-PBW algebra which is not a PBW algebra?
Can anyone provide an example of a (quadratic) Z-PBW algebra which is not a (quadratic) PBW algebra? I am using the definition of Z-PBW algebra given in Polishchuk and Positselski's book Quadratic ...
- 31
32
votes
7
answers
6k
views
Consequences of not requiring ring homomorphisms to be unital?
As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...
- 6,782
9
votes
1
answer
679
views
Mathematical software for computing in integral group rings of discrete groups?
I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...
- 2,758
5
votes
2
answers
2k
views
Sylow's theorem 3rd Proof Page 96 I.N.Herstein
I was just going through the 3rd Proof of Sylow's theorem given in the "Topics In Algebra" Book by I.N. Herstein. It looked very interesting and i really liked its Philosophy. My question what is its ...
- 4,795
3
votes
2
answers
1k
views
endomorphism ring of a finite-length module
Can anyone tell me why the endomorphism ring of a finite-length module is artinian? Bonus points if you can do it without using the radical, semisimplicity, Fitting's lemma or anything fancy. If you ...
- 3,605
18
votes
5
answers
2k
views
What is the spectrum of the ring of entire functions?
Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z_0 \in \mathbb{C}$.
...
- 3,268
7
votes
3
answers
915
views
Decidability of matrix algebra
Take multi-sorted first-order logic with equality, complex scalars, 1xn vectors, nx1 vectors, nxn matrices, addition and multiplication for each pair of sorts they make sense for, and hermitian ...
user5810
7
votes
5
answers
3k
views
Indeterminate "$x$" in algebra/ring Theory [closed]
How do you interpret the indeterminate "$x$" in ring theory from the set theory viewpoint? How do you write down $R[x]$ as a set? Is it appropriate/correct to just say that
$$R[x] = \{ f: R \to R \...
- 95
3
votes
2
answers
2k
views
Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 2,857
4
votes
0
answers
179
views
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
4
votes
2
answers
1k
views
Semiprime (but not prime) ring whose center is a domain
The center of a prime ring is a domain and the center of a semiprime ring is reduced.
Now I have no evidence to believe that if the center of a semiprime ring R is a domain,
then R has to be a ...
- 279
15
votes
1
answer
1k
views
Countable Hom/Ext implies finitely generated
Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...
- 56.6k
3
votes
2
answers
170
views
Equivariant maps of "higher order"
Given a group $G$, a ring $R$ and two $R[G]$-modules $M,N$. Then one can consider $Hom_R(M,N)$ and define inductively submodules $A_0,A_1,...$ via
$A_0:=0$
$A_{n+1}:=\{ \;f\; |\; \forall\; g\in G: (...
- 11.1k
2
votes
1
answer
628
views
Smith Normal Form and lower triangular Toeplitz Matrices
I am working on a undergrad research project with some other guys. Now the conjecture (unrelated to this question) we are trying to prove boils down to a final subproblem:
Let $A (n \times n)$ be a ...
- 21
28
votes
4
answers
5k
views
When are modules and representations not the same thing?
I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring $...
- 1,872
9
votes
2
answers
4k
views
*-homomorphisms between matrix algebras
Edited question:
Are there any other non-trivial *-homomorphisms between matrix algebras apart from the unitary homomorphisms?
Original question:
Does there exist a surjective (but not bijective) *...
- 397
1
vote
1
answer
2k
views
Cardinality of symmetric group [duplicate]
Possible Duplicate:
Cardinality of the permutations of an infinite set
Why does the symmetric group on an infinite set X have the cardinality of the power set ${\cal P}(X)$?
- 2,857
13
votes
2
answers
1k
views
Maximal ideal that annihilates entire ring
Does there exist a ring $R$ with a nonzero maximal ideal $M$ such that $R^2=R$ and $MR = RM = 0$?
Here $R$ is associative but does not have an identity (obviously). It seems a simple enough question ...
- 131
4
votes
2
answers
1k
views
Representation of rings
The endomorphisms of an abelian group form a ring under pointwise group operation and composition. Every ring is isomorphic to a subring of the endomorphism ring of some abelian group (left module ...
user5810
9
votes
2
answers
471
views
Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?
Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?
Background:
In his wonderful, wonderful book Geometric Algebra, Emil Artin describes ...
- 573
2
votes
1
answer
1k
views
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 ...
- 427
1
vote
0
answers
533
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 \...
- 11
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
20
votes
1
answer
3k
views
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 ...
- 47.3k
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:
...
- 3,612
5
votes
3
answers
797
views
Euclidean function of Euclidean domain defined at 0
In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as ...
114
votes
2
answers
12k
views
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 ...
- 4,736
39
votes
5
answers
5k
views
When does a ring surjection imply a surjection of the group of units?
The following might be a very trivial question. If so, I don't mind it being closed, but would appreciate a reference where I could read about it.
Let $R$ and $S$ be commutative rings and let $R^\...
- 33.3k
39
votes
5
answers
4k
views
Is there an explicit construction of a free coalgebra?
I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the ...
- 9,132
1
vote
1
answer
527
views
Krull Dimension
For all $n$, I need to find examples of rings $A\subset B$ such that:
i) $\dim A-\dim B\gt n$
ii) $\dim B-\dim A\gt n$
(where $\dim$ is the Krull dimension)
- 133
17
votes
2
answers
4k
views
Gaussian primes, quaternion primes, ... octonions?
Is there a notion of an octonion prime?
A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime.
A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is ...
- 151k
1
vote
2
answers
652
views
Understanding the modules of semiprimitive rings
As far as I understand, a semiprimitive ring can be fully 'explored' by its simple modules, in the sense that a semiprimitive ring is the subdirect product of its simple modules (for brevity, I'll use ...
- 2,513
3
votes
4
answers
720
views
How much are reduced powers different?
Given two infinite sets $X$ and $I$, and a filter ${\cal F}$ on $I$, one defines as usual the equivalence relation $\approx_{\cal F}$ on $X^I$ and obtains the reduced power $Y = X^I / \approx_{\cal F}$...
3
votes
2
answers
518
views
Explicit representations of finite fields
An old question that occurred to me again recently: are there any explicit formula known for sequences of irreducible polynomials $g_{p^n}(X)$ in $Z/pZ[X]$ such that for the finite field with $p^n$ ...
- 2,151
2
votes
1
answer
445
views
Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras
Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.
Question: Is it true that we can always find a positive integer $n$, a $C$-...
- 279
22
votes
6
answers
6k
views
What are hypergroups and hyperrings good for?
I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...
- 5,094
6
votes
2
answers
1k
views
Are the banded versions of a positive definite matrix positive definite?
Consider $M$, a positive definite matrix. Let $M^{(1)}$ be the diagonal matrix which agrees with $M$ on the diagonal ($M_{ii}=M^{(1)}_{ii}$). We have that $M^{(1)}$ is positive definite because it is ...
- 375