Questions tagged [ra.rings-and-algebras]
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
351
questions
32
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7
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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 ...
- 6,722
31
votes
1
answer
2k
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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 ...
- 4,525
31
votes
1
answer
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Is this formal noncommutative power series identity known?
I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...
- 109k
30
votes
3
answers
3k
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Infinite-dimensional normed division algebras
Let's say a normed division algebra is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$.
There are only four ...
- 21.5k
30
votes
7
answers
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"Sums-compact" objects = f.g. objects in categories of modules?
Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...
- 5,492
29
votes
0
answers
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Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?
Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
- 1,702
28
votes
2
answers
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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 ...
- 49.6k
26
votes
2
answers
2k
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Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Suppose we have a function $f(x_1 ,x_2 ,x_3 ,x_4).$ We know that we can factor it in two ways as $f(x_1 ,x_2 ,x_3 ,x_4)=\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Show that ...
- 519
24
votes
4
answers
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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 ...
- 3,247
24
votes
12
answers
3k
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Constructions unique up to non-unique isomorphism
1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...
24
votes
2
answers
3k
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Does any textbook take this approach to the isomorphism theorems?
Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______________, but groups will get the points ...
- 12k
24
votes
3
answers
12k
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Why is a ring called a "ring"?
Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...
- 149k
23
votes
3
answers
1k
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Is $\widehat{\mathbb{Z}}[[t]]\cong\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]$?
Let $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]] := \varprojlim_{n,m}(\mathbb{Z}/n)[x]/(x^m-1)$ be the complete group algebra of the profinite free group of rank 1. In Corollary 5.9.2 of Ribes-...
- 7,161
23
votes
2
answers
2k
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Koszul duality between Weyl and Clifford algebras?
Koszul duality
Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can ...
- 8,409
22
votes
1
answer
1k
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Codes, lattices, vertex operator algebras
At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention to ...
- 2,130
22
votes
4
answers
1k
views
What is a Kelley ring?
I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...
- 21.5k
22
votes
4
answers
2k
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Freeness of a Z[x]-module
Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...
- 19.5k
21
votes
4
answers
1k
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Can one show the equivalence of the abstract and classical Jordan decompositions for simple Lie algebras without complete reducibility?
The following fact is basic in the theory of complex Lie algebras:
Theorem. Let ${\mathfrak g} \subset {\mathfrak gl}_n({\bf C})$ be a simple Lie algebra, and let $x \in {\mathfrak g}$. Let $x = ...
- 109k
21
votes
3
answers
1k
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Nice algebraic statements independent from ZF + V=L (constructibility)
Background and motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1_\mathbb{Z}(A,\mathbb{Z}...
user1437
21
votes
10
answers
4k
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Why do we associate a graph to a ring? [closed]
I don't know if it is suitable for MathOverflow, if not please direct it to suitable sites.
I don't understand the following:
I find that there are many ways a graph is associated with an algebraic ...
21
votes
3
answers
2k
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Is there a "categorical" description of Grothendieck's algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...
- 53.1k
21
votes
1
answer
1k
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Is there any non-commutative ring such that every element other than the identity is a zero divisor?
A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
- 9,636
20
votes
2
answers
2k
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Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 2,190
19
votes
2
answers
676
views
Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...
user1437
19
votes
1
answer
740
views
$\Lambda$-Ring Structures on $\mathbb A^2$
A $\Lambda$-ring structure on a torsion-free ring over $\mathbb Z$ is a commuting family of endomorphisms $\psi_p$ satisfying $\psi_p(x) \equiv x^p$ mod $p$.
One $\Lambda$-ring structure on $\mathbb ...
- 137k
18
votes
4
answers
1k
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Bass' stable range condition for principal ideal domains
In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:
For every $n$ elements $v_1, \ldots, v_n$ that generate the unit ideal there are numbers $r_1, \ldots ...
- 245
18
votes
5
answers
2k
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Is a complete homogeneous symmetric polynomial irreducible?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h_a=\text{ sum of all monomials of degree }...
- 446
18
votes
3
answers
12k
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Künneth formula for cohomology
$\DeclareMathOperator\Hom{Hom}$Is there an algebraic Künneth formula for cohomology?
More precisely assume $A_{*}, B_{*}$ are chain complexes of free $R$-modules ($R$ is a $PID$) and $M, N$ are $R$-...
- 1,297
17
votes
1
answer
2k
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Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.2k
17
votes
0
answers
687
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,200
16
votes
1
answer
1k
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Two rings...are they isomorphic?
Edit: I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments.
Consider the local rings
$$R = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz\...
- 2,527
16
votes
3
answers
1k
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Two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers
I am searching for two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers. Who knows of such examples? Thanks a lot.
- 161
15
votes
2
answers
1k
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Are there only finitely many associative algebras of fixed dimension?
Given an algebraically closed field $F$, for any positive integer $n$, are there always only finitely many non-isomorphic (noncommutative) associative algebras (possibly without identity) with ...
- 704
15
votes
1
answer
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How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras?
There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.
What about just traces on separate algebras? That is, take one of them,...
- 17.5k
15
votes
3
answers
3k
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Are there other semidirect product/crossed products in other areas
Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only ...
- 3,954
15
votes
2
answers
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When is bar-cobar duality an equivalence?
Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
- 4,083
15
votes
2
answers
1k
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Exact sequence of monoids
What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
http://www.math.ucla.edu/~balmer/Pubfile/...
- 2,949
15
votes
1
answer
2k
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Automorphisms of $P(\Bbb N)$
I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the ...
- 1,435
15
votes
1
answer
1k
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Are wild problems related to undecidable ones?
In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...
- 5,594
14
votes
1
answer
4k
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Do these matrix rings have non-zero elements that are neither units nor zero divisors?
First, a disclaimer: This is a repost of a question I asked on stackexchange (no answer there).
Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with ...
- 1,187
14
votes
3
answers
610
views
Polynomial Rings
Let $R$ and $S$ be non-zero rings with identity. Is it possible to have $R[x] \cong S[[x]]$ ?
user30230
14
votes
2
answers
1k
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Obstructions for a group to be the multiplicative group of a field [duplicate]
It is well known that every finite multiplicative subgroup of a field is cyclic.
I somehow got interested in a possible reverse implication:
Assume we have an abelian group $G$ whose every finite ...
- 6,621
13
votes
2
answers
646
views
Noncontractible connected topological rings ?
Are there any non-contractible connected topological rings?
Of course, such a thing cannot be a (topological) algebra over the reals.
(I have a vague memory of having a glance at an erticle by Lurie ...
- 22.7k
13
votes
4
answers
2k
views
What is about J. v. Neumann's "Continuous geometry"?
I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of ...
- 10.7k
13
votes
1
answer
329
views
Existence of a translation-invariant basis of $\ell^2$
This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural).
I give four equivalent formulations of the same ...
- 30.2k
13
votes
1
answer
1k
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Why not _co_free modules?
Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...
- 53.1k
13
votes
5
answers
3k
views
Noncommutative localization of a ring: complete construction
I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases.
Let $R$ be a non-commutative ring and $S$ a ...
- 365
13
votes
1
answer
776
views
Is $\mathbb{Q}_p \otimes_{\mathbb{Q}}\mathbb{Q}_p $ coherent?
Let $\mathbb{Q}_p$ denote the field of fractions of $\mathbb{Z}_p$. By the answers to this quesition the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ cannot be a Noetherian ring (...
- 3,655
13
votes
3
answers
356
views
How should one look at the set of compatible ring structures on a given group?
Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, ...
- 19.2k
12
votes
4
answers
2k
views
Subalgebras of matrices
Given a matrix algebra over a field, can one describe all its subalgebras?
- 121