# 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.

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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 ...

**7**

votes

**1**answer

280 views

### Does this algebra have finite global dimension ? (Human vs computer)

Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...

**2**

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81 views

### A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...

**3**

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52 views

### Explicit examples of finite dimensional, involutive Hopf algebras with traceless antipode?

$\require{AMScd}$
In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless.
Unfortunately, the method of proof in [...

**8**

votes

**1**answer

240 views

### Description of p-adics tensor the reals

What is $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{R}$ equivalent to?
where $\mathbb{Z}_p$ are the p-adic integers.
I am specially interested in the case $p=2$.
Do know that $\mathbb{Z}_p\otimes_{\...

**5**

votes

**1**answer

140 views

### $Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$

$G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have
$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but ...

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39 views

### tensor product of two compact induced representations

Suppose $A \subset A^{\prime}$ and $B \subset B^{\prime}$ are all p-adic groups, and $V_{\pi}$ is a representation of $A$; $V_{\rho}$ is a representation of $B$.
Define
$ind_{A}^{A^{\prime}}V_{\pi}...

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58 views

### Algebraic description of the reduced incidence algebra of a poset

In the book "Combinatorial theory" by Martin Aigner (from 1979), the standard algebra of a poset is introduced as the subalgebra of the incidence algebra of a poset consisting of the functions that ...

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111 views

### How to formulate supercommutativity in a characteristic free way?

I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...

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145 views

### Slightly noncommutative Nakayama's lemma?

Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...

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**1**answer

96 views

### Finite generation of flat deformations of algebras

Let $R=\mathbb C[q^{\pm 1}]$ and let $A$ be a graded (possibly non-commutative) $R$-algebra, $A=\oplus_{n=0}^\infty A_n,$ where $A_0=R$ and all $A_n$'s are free $R$-modules.
Then $A'=A/(q-1)$ is a ...

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91 views

### Has the “semidirect monoid of a semiring” been considered anywhere?

Given a semiring $S$, we get a monoid $M(S)$ as follows:
The underlying set of $S$ is $S^2$
The identity element is $(0,1)$
The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...

**29**

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**1**answer

2k views

### Have you ever seen this bizarre commutative algebra?

I have encountered very strange commutative nonassociative algebras without unit, over a characteristic zero field, and I cannot figure out where do they belong. Has anybody seen these animals in any ...

**4**

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**1**answer

148 views

### Is the category of rational Lie algebras monoidal?

I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes_{\mathbb{Q}} B$ is a rational ...

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**0**answers

46 views

### Algebra dimension computation in GAP

How does GAP compute the dimension of a matrix algebra over the rational numbers? I am curious about the run time.
For example, the manual https://www.gap-system.org/Manuals/doc/ref/chap62.html does ...

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votes

**2**answers

216 views

### Ideals invariant under ring automorphisms

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...

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546 views

### A new generalization of the dimension?

During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...

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**0**answers

65 views

### Non-zero homomorphism from a module to its ground ring

Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\...

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134 views

### Rings that fail to satisfy the strong rank condition

In T.Y. Lam's book Lectures on Modules and Rings, a ring $R$ is said to satisfy the strong rank condition if, for every natural number $n$, there is no right $R$-module monomorphism $R^{n+1}\to R^n$. ...

**4**

votes

**1**answer

125 views

### Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$.
Question 1: Is there a simple proof that $...

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votes

**1**answer

128 views

### Global dimension of a graded algebra

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...

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votes

**1**answer

201 views

### Quadratic algebras and Koszul algebras

Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$.
In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...

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23 views

### Defect of subnormality in unit groups of modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. $G$ is normal in $1+rad(KG)$ if and ...

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**1**answer

103 views

### “Solution” of finite cluster algebras

Consider the cluster algebras $A_n$ and $D_n$. Choose any cluster $x$, is there an explicit formula that express all other cluster variables in terms of $x$?

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**1**answer

154 views

### Representations of degenerate Clifford algebras

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...

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76 views

### “Cross-Ratios” for D_n cluster algebra

Both cluster algebras $A_n$ and $D_n$ admit an interpretation in terms of (tagged) triangulations of Riemann surfaces - respectively a Poincaré disk with n+3 punctures on the boundary and a Poincaré ...

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**1**answer

85 views

### Does this element belong to all powers of the augmentation ideal of the group algebra.

Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $...

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**1**answer

314 views

### Decide if a matrix is transposable

A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...

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votes

**2**answers

367 views

### Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$

For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.
For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...

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109 views

### action of Weyl group element on Weyl vector

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...

**10**

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**1**answer

401 views

### Functoriality of the Hopf dual

Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...

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60 views

### About crossed product of the group von Neumann algebra

Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...

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42 views

### Quotient of quasi-isomorphic nonpositively graded cdga's

I'm looking for a theorem about quotient of quasi-isomorphic cdga's:
Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) concentrated in nonpositive degree, and $\mathfrak ...

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97 views

### Projective modules over maximal orders of central simple algebras

In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let $R ...

**4**

votes

**1**answer

120 views

### Quotient of quasi-isomorphic cdga's

I'm looking for a theorem about quotient of quasi-isomorphic cdga's:
Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) of nonpositive degrees, and $\mathfrak m \subset A, ...

**3**

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**0**answers

75 views

### Jacobson radical of a tensor product

Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...

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38 views

### $c$-matrix reduction in hereditary algebras

Let $k$ be an algebraically closed field, $Q$ be a finite connected quiver and $Q'$ a subquiver of $Q$. Let $C$ be the $c$-matrix of a chamber of the scattering diagram/semi-invariant picture of $kQ$. ...

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**1**answer

120 views

### If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...

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61 views

### $\Omega^2(S) \cong \tau(S)$ for simple modules

Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra?
Here $\tau$ denotes the Auslander-Reiten translate, which is ...

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votes

**1**answer

232 views

### A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...

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119 views

### Cellular and primary binomial ideals

Let $I \subseteq \mathbb{K}[x_1, \dots, x_n]$ be an ideal of a polynomial ring over a field $\mathbb{K}$.
$I$ is called cellular if every variable $x_i$, with $i=1, \dots, n$, is either a ...

**8**

votes

**1**answer

94 views

### Injective indecomposable modules over Laurent polynomial rings

What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...

**6**

votes

**1**answer

193 views

### Zero divisors in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...

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62 views

### Characterisation of projective modules over tensor products of fields

Let $k$ be a field and $L_1$ and $L_2$ finite field extensions of $k$.
Let $A:=L_1 \otimes_k L_2$ as an algebra.
Question:
Given a finitely generated $A$-module $M$, do we have that $M$ is ...

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101 views

### Unitary element of the group algebra

Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...

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86 views

### Does this element belong to $\mathbb CG$?

Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...

**4**

votes

**2**answers

296 views

### When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...

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**0**answers

83 views

### Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...

**5**

votes

**2**answers

181 views

### Naturality of PD model of a CDGA

In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are ...

**3**

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**0**answers

60 views

### conditions for algebra cohomology finiteness

Let $A=\bigoplus\limits_{n\geq 0}A_n$ be a graded (unital associative) ring, say an algebra over a field with finite dimensional graded components and $A_0$ semisimple. Are there reasonable conditions ...