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.

Filter by
Sorted by
Tagged with
0
votes
0answers
33 views

Flat modules and subrings

Let $R$ be ring and $S \subseteq R$ a subring. Can it happen that there exists a flat $R$-module $M$ that is not flat as an $S$ module? Does anything change if we replace flat by faithfully flat?
1
vote
1answer
50 views

Simplicial set construction of the classifying space

What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?: Regard $G$ as a category with ...
0
votes
0answers
59 views

Fundamental theorem of algebra for sedenions

Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.² Does similar ...
6
votes
2answers
287 views

Is the triple product in a Freudenthal triple system fully symmetric?

I'm trying to learn about Freudenthal triple systems. Here is the definition given by Helenius [1], start of Section 5: A Freudenthal triple system is a finite-dimensional vector space $V$ over a ...
3
votes
2answers
127 views

On submodules of vector fields

I don't know much about modules aside from their basic definition and that they are more complicated than vector spaces. I am asking this question because I wish to have a more "algebraic" ...
2
votes
0answers
113 views

Actions of rings (and other algebraic structures) on abelian categories

On the project I am currently working on, there are abelian, Krull-Schmidt categories $\mathcal{C}$ where it seems natural to equip $\mathcal{C}$ with the action of a ring $R$ (in some cases a ...
1
vote
0answers
73 views
+50

Nilpotent elements of index $2$ in group algebra $FA_4$

Let $A_4 = K_4 \rtimes C_3$ be alternating group on $4$ symbols and $F$ be finite field containing $4$ elements. By definitions of group algebra and augmentation ideal, there exist a natural map $$\...
6
votes
2answers
188 views

Different Bialgebra/Hopf algebra structures on coalgebras

Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
1
vote
0answers
84 views

Finding an injective envelope containing another injective envelope

Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
0
votes
0answers
26 views

Derivations of n-th Weyl algebra

I have known that derivations on the first Weyl algebra are all inner. However,for the higher order algebras,does all derivatioms are inner ?
2
votes
0answers
179 views

What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?

This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms \begin{align*} \mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...
6
votes
1answer
954 views

Solid rings and Tor

A solid ring is a ring $R$ such that the multiplication $R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism. These were classified by Bousfield and Kan; they are subrings of $\mathbb{Q}$, $\mathbb{Z}/...
3
votes
1answer
115 views

Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
4
votes
0answers
119 views

Does the tropical semiring admit a universal property?

Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
1
vote
0answers
46 views

Indecomposable comodules

For a Hopf algebra $A$, we say that a comodule $V$ is indecomposable if it is not equivalent to a direct sum of irreducible comodules. $\bullet$ What is an example of a finite dimensional ...
2
votes
1answer
408 views

What is a coalgebra?

A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...
4
votes
0answers
72 views

Associative rings with "big" commutative subrings

Let $A$ be an associative ring and $R\subset A$ be a commutative subring. Suppose that every element of $A$ has the form $urv$ where $r\in R$ and $u, v\in A^*$ are invertible. A basic example is $A=...
3
votes
0answers
263 views

Homology $H_{\ast}(T, V)$

Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$; i.e. $V:=\left\{\left( ...
5
votes
0answers
137 views

Generalising supercommutativity as a grading by the $1$-truncated sphere spectrum

A discussion that has been going recently is that supersymmetry corresponds to grading over the sphere spectrum, coming from an insight due to Kapranov. To formalise such a statement, one needs a ...
8
votes
0answers
290 views

In search of lost graded rings

$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
1
vote
1answer
163 views

A closed subgroup $G$ of $\operatorname{GL}_2 \mathbb{Z}_\ell$ which surjects onto $\operatorname{GL}_2 \mathbb{F}_\ell$

Let $\ell \ge 5$ be a prime and $G$ be a closed subgroup of $\operatorname{GL}_2 \mathbb{Z}_\ell$ whose image in $\operatorname{GL}_2 \mathbb{F}_\ell$ is $\operatorname{GL}_2 \mathbb{F}_\ell$. Then $G ...
7
votes
1answer
376 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. ...
2
votes
0answers
96 views

Koszul differential of the complex $\bigwedge \mathfrak{g}^*$

Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The definition of the Koszul differential is given in the article by Kumar and Vergne on equivariant cohomology page 133 as follow: Let us now ...
5
votes
1answer
2k views

Are all topological (finite-dim) real vector spaces homeomorphic to a coordinate space?

I know that all real, finite-dimensional topological vector spaces are isomorphic to $\mathbb{R}^n$ for some $n$, but are they also homeomorphic? The reason I'm asking this is because I was wondering ...
48
votes
7answers
12k views

Good lattice theory books?

A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
4
votes
1answer
137 views

A weight generalization of root systems?

For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...
2
votes
0answers
101 views

Hypermodulus and what mathematical objects have it

When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ...
30
votes
15answers
11k views

Geometrical meaning of Grassmann algebra

I don't understand wedge product and Grassmann algebra. However, I heard that these concepts are obvious when you understand the geometrical intuition behind them. Can you give this geometrical ...
3
votes
1answer
129 views

The going-up theorem for free extensions of almost commutative rings

I would like to know whether or not the going-up property holds for some classes of finite filtered extensions of non-commutative rings. Let $S \subseteq R$ be rings. The pair $(S,R)$ has the going-...
2
votes
0answers
37 views

Decomposition of augmentation ideal in a group ring

Let $R$ be a ring and $G$ be a finite group with invertible order in $R$. Assume that the augmentation ideal $\Delta (G)$ of group ring $RG$ has a decomposition $M_1\oplus M_2\oplus \cdots M_k$ as an $...
2
votes
0answers
98 views

Extending elements of the dual of a Hopf subalgebra to the Hopf dual of the whole algebra

I have a question about Hopf duals. To begin we can remind the definition: For an Hopf algebra $A$ over a field $k$, the Hopf dual $A^{\circ}$ is the subspace of the linear dual $\mathrm{Lin}_k(A,k)$ ...
10
votes
1answer
207 views

Plane partitions as irreducible representations

The irreducible representations of the symmetric group algebras $A_n=KS_n$ over a the complex numbers (or any field of characteristic 0) $K$ satisfy the following properties: The irreducible ...
9
votes
1answer
234 views

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 ...
4
votes
0answers
179 views

Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
4
votes
0answers
110 views

Are PD-, $\lambda$-, $\psi$-, and $\delta$-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 ...
3
votes
0answers
75 views

Efficient computation of "higher order" Jacobi symbols

Let $N> 1$, and suppose $a \in (\mathbb{Z}/N\mathbb{Z})^{\times}$. There are efficient algorithms to compute the Jacobi symbol $\left(\frac{a}{N}\right)$ using quadratic reciprocity. Fix $k \geq 1$....
1
vote
1answer
112 views

Using equational Jacobson condition to prove element lies in radical of ideal

Recall the Jacobson radical of a ring consists of elements $f\in A$ such that $1-gf\in A^\times$ for every $g\in A$. Say an ideal $I\vartriangleleft A$ is Jacobson if in the quotient $A/I$ the ...
15
votes
1answer
376 views

Defining Massey products as transgressions

Let $A$ be a dg algebra, and $x, z \in A$ cocycles. Let's consider the maps $$ A \to A \oplus A \to A$$ given by $y \mapsto (xy,yz)$ and $(u,v) \mapsto uz-xv$, respectively. We think of this as ...
5
votes
1answer
167 views

What is a Serre-smooth algebra?

Let $A$ be an $R$-algebra. In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction. But no formal ...
2
votes
0answers
74 views

Minimizing the spectral radius of certain elements of group rings

Let $G$ be a finite group. Let $I_{G}$ be the ideal on the group ring $\mathbb{C}[G]$ consisting of elements of the form $\alpha\cdot\sum_{g\in G}g$. Let $\lambda_{n}(G)$ be the minimum spectral ...
5
votes
2answers
279 views

Classifying Hopf algebras that admit a single irreducible comodule

Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule $$ k \to k \otimes H, ~~ k \mapsto k ...
2
votes
3answers
330 views

The existence of two maximal ideals with the same set of idempotents

Let $R$ be a commutative ring with identity and $A$ and $B$ be two proper ideals of $R$ such that $A+B=R$ and for each $r^2=r\in R$ we have either $r\not\in A$ or $r-1\not\in B$. How can we prove the ...
26
votes
2answers
1k views

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 ...
3
votes
0answers
116 views

Does the category of rings embed fully faithfully into the category of $\mathbb{F}_{1}$-algebras?

The idea of a theory of algebraic geometry over the "field with one element" $\mathbb{F}_{1}$ is to give a fully faithfully embedding of categories $$\mathsf{Sch}_{\mathbb{Z}}\hookrightarrow\...
12
votes
2answers
936 views

Subsets of the integers which are closed under multiplication

Let $S$ be a subset of the integers which is closed under multiplication. There are many possible choices of $S$: $S = \{-1, 1\}$. $S$ is the set of integers of the form $a^k$, where $a$ is fixed and ...
8
votes
1answer
212 views

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 $S^n$ ...
14
votes
1answer
480 views

Group rings such that every (countably generated) module has a maximal submodule

Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma. I am interested in the following question, with two variants. ...
5
votes
2answers
247 views

Is this quiver with relations of finite representation type

Let $Q=(Q_0,Q_1)$ be the following quiver, $Q_0$ consist of 2 vertices, denoted by 1,2. $Q_1$ consist a loop at 1 called $\gamma$, an arrow $\alpha$ from 1 to 2 and an arrow $\beta$ from 2 to 1. The ...
2
votes
0answers
50 views

Wedderburn decomposition of wreath product of cyclic p-groups

Let $G$ be wreath product of cyclic group of prime order $p$ by itself, i.e. $G=C_p \wr C_p$, where the action of $C_p$ is taken as cyclic permutation on generators of first $p$ cyclic groups. Can we ...
1
vote
0answers
60 views

Has anyone studied this possible generalisation of the Singular Value Decomposition to all commutative $*$-rings?

I don't know any abstract algebraists personally, which is why I'm asking this question here. Let $(R,*)$ be a commutative $*$-ring where $*:R \to R$ is an involution. Each conjecture below is stated ...

1
2 3 4 5
57