All Questions
Tagged with rings-and-algebras or ra.rings-and-algebras
3,500 questions
0
votes
1
answer
460
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Sp(2n) intersect Sp(2n,H)? (Please read for explanation of notation)
First let me fix some notation:
Let $O(n)$ be the group of $n \times n$ real matrices $T$ which are "orthogonal", $U(n)$ be the group of $n \times n$ complex matrices $T$ which are "unitary" and $Sp(...
- 177
8
votes
3
answers
2k
views
Rational exponential expressions
Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...
- 2,283
16
votes
2
answers
1k
views
Which commutative groups are the group of units of some field?
Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...
- 118k
5
votes
1
answer
551
views
What is an exponential?
Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on
tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential ...
- 15.4k
16
votes
6
answers
2k
views
What is an algebraic group over a noncommutative ring?
Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I know what "linear ...
- 54.6k
31
votes
11
answers
10k
views
Introduction to deformation theory (of algebras)?
So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
- 12.6k
43
votes
3
answers
7k
views
transcendental Galois theory
Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
- 65.4k
7
votes
1
answer
732
views
Does ⬦ generate all De Morgan algebras?
(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...
- 1,119
1
vote
3
answers
474
views
Fractional powers of Dirichlet series?
Let $R$ be the ring of Dirichlet series with integer coefficients. I'd often wondered about whether $R$ was a UFD; this post cleared that up, because it turns out that $R\simeq\mathbb{Z}[[x_1,x_2,\...
- 6,782
3
votes
2
answers
611
views
Computing Integral Closures
I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of &#...
- 1,386
2
votes
2
answers
295
views
References for traceless and/or imaginary Octonionic matrices?
Hi all.
I was wondering if anyone has seen any work related to either traceless matrices of Octonions (with trace defined as the sum of diagonal) or matrices of pure imaginary Octonions (meaning real ...
- 177
4
votes
2
answers
322
views
A hands-on description of a "completion" of the free commutative monoid on countably many generators
This is basically an I'm-weak-at-algebraic-geometry question. I asked it as a warm-up question here, but Ilya N asked me to break that post up into several questions.
Consider the free commutative ...
- 54.6k
9
votes
3
answers
987
views
Octonionic Unitary Group?
Hi all.
I was wondering if anyone has any references on work related to the Octonionic Unitary group. I would imagine that such a group would be generated by Octonionic skew-Hermitian matrices (at ...
- 177
2
votes
1
answer
630
views
Name for semiring with weakened annihilation law?
The axioms of a semiring include:
$$0·a = 0 = a·0$$
Is there a name for an algebraic structure which satisfies all the axioms of a semiring except for this one?
(if it helps, the structure I have ...
- 3,267
11
votes
2
answers
1k
views
What are important examples of filtered/graded rings in physics?
Hi,
what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics?
Of ...
- 13.2k
4
votes
0
answers
534
views
Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?
A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H \rightarrow G$, and $\alpha: G \times H \rightarrow H$ that defines an action of $G$ on $H$, $\tilde{\alpha}$: ...
- 5,264
24
votes
6
answers
7k
views
Introduction to W-Algebras/Why W-algebras?
Does anyone know of an introduction and motivation for W-algebras?
Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice ...
- 13.2k
24
votes
4
answers
3k
views
does the "convolution theorem" apply to weaker algebraic structures?
The Convolution Theorem is often exploited to compute the convolution of two sequences efficiently: take the (discrete) Fourier transform of each sequence, multiply them, and then perform the inverse ...
- 3,267
19
votes
2
answers
7k
views
Generalization of the shakehands/condom puzzle?
The classic handshake puzzle goes something like this:
"Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?"
Its common ...
- 235
87
votes
5
answers
10k
views
When is $A$ isomorphic to $A^3$?
This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...
- 63.1k
5
votes
1
answer
572
views
Non-smooth algebra with smooth representation variety
A not necessarily commutative algebra A (over C, say) is called formally smooth (or quasi-free) if, given any map $f:A \to B/I$, where $I \subset B$ is a nilpotent ideal, there is a lifting $F:A \to B$...
- 2,869
12
votes
5
answers
5k
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reduced ⊗ reduced = reduced; what about connected?
Several questions actually.
All rings and algebras are supposed to be commutative and with $1$ here.
(1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $...
- 33.8k
7
votes
1
answer
567
views
Depth Zero Ideals in the Homogenized Weyl Algebra
Let $\mathcal{D}$ be the $n$th Weyl algebra $ \mathcal{D} :=k[x_1,...,x_n,\partial_1,...,\partial_n] $, where $\partial_ix_i-x_i\partial_i=1$.
Let $\widetilde{\mathcal{D}}$ be its Rees algebra, ...
- 13k
2
votes
1
answer
160
views
Lower bound for characteristic variety
Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.
Does the following then hold?
dim Ch(M) \...
- 23
10
votes
1
answer
835
views
what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
user2529
3
votes
2
answers
889
views
computation, algebra, logic
So a really simple way of describing a digital computer is to say that it is a device for performing boolean operations. You feed it a bunch of bit strings, which is a description of the problem and ...
user577
4
votes
1
answer
182
views
When is the essential extension commutes with colimits(or push forward)
Let $M$ be an $R$-module,where $R$ is a hereditary (or cohomological dimension less or equal to 1).Take $E(R)$ to be injective hull of $R$, then we have the essential extension
$i:R^I\rightarrowtail ...
- 5,515
10
votes
2
answers
1k
views
When is every submodule pure?
Recall that a module is called
semisimple if every submodule is a direct summand
pure semisimple if every pure submodule is a direct summand
There is quite a bit of work on semisimple and pure ...
- 3,685
24
votes
1
answer
2k
views
Sums of injective modules, products of projective modules?
Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
Analogously, under what assumptions on R does a ...
- 15.6k
15
votes
3
answers
3k
views
Which is the correct universal enveloping algebra in positive characteristic?
This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let $\...
- 54.6k
12
votes
2
answers
2k
views
Topological Rings
Is it true that, if S is a subring of a separable topological Noetherian ring R,
then S is separable, too ?
- 4,060
95
votes
11
answers
6k
views
Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
- 2,600
6
votes
1
answer
875
views
What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?
What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...
- 4,593
13
votes
1
answer
5k
views
What are tame and wild hereditary algebras?
What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
- 2,992
5
votes
4
answers
6k
views
Are quotients of polynomial rings almost UFDs?
If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\...
- 1,412
96
votes
16
answers
18k
views
Why is it a good idea to study a ring by studying its modules?
This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
- 118k
4
votes
1
answer
135
views
(in-)compatible gradings of an associative algebra tell us...?
If an associative algebra A is $\mathbb{Z}$-graded, then it is automatically $\mathbb{Z}\_2$ (aka $\mathbb{Z}/2\mathbb{Z}$) graded by defining $A\_{\bar{0}}$ to be ...
- 411
0
votes
3
answers
469
views
Explanation and Definition of Iwahori order
Can anyone explain what Iwahori order is? All I know is that it is mentioned here.
- 386
9
votes
5
answers
2k
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Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring
Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?
- 386
6
votes
3
answers
4k
views
Intuitive Example of a Jacobson Radical
Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand Wikipedia's explanation.
- 386
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...
25
votes
8
answers
6k
views
What is the "right" definition of a ring?
This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object ...
- 118k
19
votes
2
answers
1k
views
Hopf algebra reference
I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
- 156k
8
votes
1
answer
451
views
Separable and finitely generated projective but not Frobenius?
Let R be a commutative ring, and $A$ an $R$-algebra (possibly non-commutative). Then $A$ is separable if it is finitely generated (f.g.) projective as an $(A \otimes_R A^{\mathrm{op}})$-algebra. ...
- 27.5k
62
votes
5
answers
10k
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Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 11.2k
17
votes
3
answers
1k
views
R2 and S3 for rings.
For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition ...
- 273
8
votes
2
answers
2k
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What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would ...
- 7,237
36
votes
4
answers
5k
views
What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
- 10.5k
39
votes
4
answers
5k
views
Is there a universal property for Witt vectors?
Do the Witt vectors satisfy a universal property?
- 7,994
8
votes
1
answer
669
views
Is there a good version of Artin-Wedderburn for semisimple algebra objects?
Artin-Wederburn says that if you have a semisimple algebra then it is a product of matrix algebras over division rings.
Suppose that $C$ is a fusion category over the complex numbers (if you want to ...
- 28.1k