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.
367 questions
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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 ...
- 10.8k
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1
answer
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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 ...
- 32.5k
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3
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Classification of commutative Frobenius algebras
Are there attempts to classify commutative finite dimensional Frobenius algebras? They appear often in mathematics, such as in algebraic geometry and the famous category equivalence between ...
- 26.5k
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3
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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, ...
- 21.2k
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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 ...
- 465
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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 (...
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2
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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 ...
- 23.3k
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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 ...
- 54.6k
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Zero divisor conjecture for finite fields
I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let ...
- 4,705
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2
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Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...
- 121
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Square of primary ideals
Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
- 131
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PID expressed as finite union of subrings
There is a classical theorem that no field can be expressed as finite union of proper subfields.
In contrast, there is an example of an integral domain that can be expressed as finite union of proper ...
- 365
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1
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Relations between coefficients of expansions of a rational function at 0 and infinity
This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up."
Given two Laurent power series $A(t)=\sum_{k>N}a_kt^k$ and $B(t)=\sum_{k>M}...
- 44.7k
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Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)?
We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra $U(\mathfrak{g})$, my question is: is there any "quantum Lie algebra" embeded into the quantum enveloping ...
- 131
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3
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What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
- 2,464
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0
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Detecting invertible elements in group rings by their images for finite quotients of the group
Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent).
Consider an element ...
11
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1
answer
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PBW theorem over a Q-algebra, without freeness or flatness
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...
- 33.8k
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2
answers
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Generalizing the Fundamental Theorem of Symmetric Polynomials
The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
- 2,356
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5
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On the notion of partial semigroup
A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = (M,...
- 10.5k
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Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
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3
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Is every field extension of an ultrafield an ultrafield?
Let $K=\lim(K_{i})$ be an ultrafield (over a non-principal ultrafilter), and let $K\hookrightarrow K'$ be a field extension of $K$.
When the field $K'$ is finite over $K$ it is also an ultrafield by ...
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Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
- 716
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Is there a way to define a prime ideal object via diagrams in the category of rings?
I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...
- 30.3k
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2
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Generalization of finitely generated, finitely presented modules?
Let $R$ be a commutative ring and $M$ an $R$-module.
The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$.
Similarly, $M$ is finitely presented iff there is an ...
- 523
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Existence of a finite extension of ℤ providing a finite extension of the primes
Let $R$ be a ring (possibly noncommutative with zero-divisors). A non-unit and non-zero-divisor element $r \in R$ will be called irreducible if for all $a,b \in R$ such that $r=ab$, then $a$ or $b$ is ...
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2
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Countable Maximal Ideals
This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand (as a left ideal).
user38138
10
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Tensor products of $\infty$-algebras over operads
Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to ...
- 40.2k
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A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids
In Lemma 5.2(a) of Z. Fiedorowicz, Classifying Spaces of Topological Monoids and Categories American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
...
- 38.6k
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Are local finite dimensional Hopf algebras symmetric?
Recall that a finite dimensional algebra $A$ over a field $K$is called a Frobenius algebra in case $A \cong D(A)$ as right modules, where $D(A) \cong Hom_K(A,K)$. In case $A \cong D(A)$ as bimodules, ...
- 26.5k
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3
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On similar matrices and polynomial matrices
I'm teaching linear algebra and I'm encountering this theorem:
two matrices $A$ and B are similar iff $tI - A$ and $tI - B$ are equivalent (as polynomial matrices), where $I$ is the unit matrix.
The ...
- 357
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5
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Schroeder-Bernstein for Rings
Suppose $f: A \to B$ and $g: B \to A$ are injections of rings
(commutative with identity). Must $A$ and $B$ be isomorphic as
rings?
According to this
question, this answer should be "no", but can ...
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Are differential rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
- 11.8k
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Localization and intersection
It is very well known that if $\mathfrak p_1, \ldots,\mathfrak p_n$ are prime ideal of an integral domain $A$, then we have the equality$$S^{-1}A=\bigcap_{i=1}^n A_{\mathfrak{p}_i},$$ where $S:=A\...
- 1,252
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Which group algebras in analysis are "true group algebras"?
Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that
$$
\pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...
- 7,426
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2
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Algebraic power series of finite order
Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
- 24.2k
9
votes
2
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513
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Where should I search for resolutions?
In my research, to test some conjectures or just to illustrate some facts, I often need to compute some explicit examples of derived functors (in the sence of Quillen's model categories). Mainly I ...
- 1,276
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1
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Units of group algebra of dihedral group
Question:
Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite ...
- 63.9k
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First-order UFD (factorial ring) condition / pre-Schreier rings
All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
- 33.8k
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2
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Birkhoff's completeness theorem put into practice
Birkhoff's completeness theorem (see here, Theorem 14.19) states that an equation which is true in all models of an algebraic theory can be proven in equational logic.
Question. Does the proof of ...
- 63.1k
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1
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Generalizing detropicalization
Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
- 19.7k
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2
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Extensions of an infinite product of copies of Z by Z
The question is simple:
Let $P$ be an infinite direct product of copies of $\mathbb Z$. Do there exist any nontrivial extensions
$$0 \to \mathbb Z \to E \to P \to 0$$
in the category of commutative ...
- 4,015
8
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1
answer
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Functions over monoids which factor in two different ways
This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
- 32.1k
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2
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Polynomial roots in the ring extension
Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
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Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
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1
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Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
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1
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Density of extended Mersenne numbers?
Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in ...
- 763
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2
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Reason to apply the Koszul sign rule everywhere in graded contexts
The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded-commutative algebra generated by $n$ elements of ...
- 499
8
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2
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280
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Graded rings with compatible S_n actions
Does the following mathematical gadget have a standard name? Let $R$ be an $\mathbb{N}$-graded ring together with an $S_n$ action on each $R_n$ which are compatible in the following sense. Let $i:...
- 28.1k
8
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2
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Left-right non-bimodule examples
Let $A$ be a unital algebra, defined over the complex numbers. Any bimodule $M$ over $A$ must, by definition, be a left, and right, module satisfing
$$
a.(m.b) = (a.m).b, ~~~~~~~ \textrm{ for all } a,...
8
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3
answers
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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