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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.

2
votes
0answers
39 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 ...
4
votes
2answers
188 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}...
1
vote
0answers
380 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 ...
1
vote
0answers
56 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 $\...
8
votes
0answers
86 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
1answer
115 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 $...
3
votes
1answer
119 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(...
0
votes
0answers
39 views

Linear transformation group of paraboloid in $\mathbb{R}^d.$ [on hold]

Suppose $P$ is a paraboloid in $\mathbb{R}^d.$ What is the group of all linear transformations $\mathcal{L}: \mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $\mathcal{L}(P)=P?$ Could someone please ...
3
votes
1answer
193 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 ...
-1
votes
0answers
50 views

About Module and Modulo [duplicate]

I come across these two words frequently in Algebra.They are almost the same apart from the last letter. I know the definitions or meanings of them in mathematics totally, as I'm majoring in Algebra. ...
-1
votes
0answers
98 views

Is this basis a Schauder basis?

Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that $\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent, $(...
1
vote
0answers
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 ...
3
votes
1answer
99 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 $x$?
9
votes
1answer
151 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 ...
8
votes
0answers
73 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 ...
1
vote
1answer
83 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 $...
11
votes
1answer
308 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 ...
3
votes
2answers
363 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 $\mathrm{GL}_n(\mathbb C)$...
2
votes
0answers
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
votes
1answer
393 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 ...
1
vote
0answers
57 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 ...
3
votes
0answers
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 ...
4
votes
0answers
91 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
1answer
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
votes
0answers
72 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 ${\rm Rad}(A_i )$, $i=1,2$...
2
votes
0answers
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 $kQ$. ...
3
votes
1answer
116 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+...
3
votes
0answers
60 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 ...
2
votes
1answer
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+...
8
votes
0answers
118 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
1answer
91 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
1answer
189 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 $G$ ...
1
vote
0answers
59 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 ...
1
vote
0answers
99 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 ...
3
votes
0answers
84 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
2answers
293 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{...
3
votes
0answers
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
2answers
178 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
votes
0answers
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 ...
5
votes
0answers
87 views

Duality between coalgebras and (pseudocompact) algebras - uniqueness

The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below. Let $k$ be a field. The ...
12
votes
2answers
419 views

Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?

The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, ...
2
votes
0answers
40 views

Annihilation in algebras derived from a potential

Let $k$ be a field and $w$ be a homogeneous (may be twisted) potential of degree $s+1\ge 4$ in the algebra $k<x_1,\dots,x_n>$. This means that $w=\sum\limits_{i=1}^nx_if_i=\sum\limits_{i=1}^...
5
votes
0answers
52 views

von Neumann regular ring homomorphisms

Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat. In particular, $\mathrm{id}...
5
votes
0answers
150 views

Are these element in a group algebra of a torsion-free group zero divisors?

Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements‌ can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)? $$1+x+y,\quad 4+x+x^{-1}+y+...
5
votes
1answer
99 views

Equivalence of idempotents and projective modules over nonunital rings

For a nonunital ring $R$ (or "rng") one has to be a little bit careful when considering the category of (left or right) $R$-module and, furthermore, the standard equivalent definitions of projective ...
1
vote
0answers
64 views

Extension of a derivation

Let I be a closed left ideal of a Banach algebra A and let D:I\to I* be a derivation. Does D extend to a derivation from A to A*?
0
votes
0answers
96 views

Krull dimensions and regular sequences

I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting: Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\...
3
votes
0answers
63 views

Can we abelianize quasi-isomorphisms of dga?

Suppose that $A \leftarrow X_1 \to \dots \leftarrow X_m \to B$ is quiso of differential graded algebras, and $A, B$ happen to be (graded) commutative. Can we find commutative $Y_1, \dots, Y_n$ ...
1
vote
0answers
46 views

Rings with flat injective envelope and global dimension at most one

Is there a classification of rings $R$ with the following properties: -The injective envelope of $R$ is flat. -The global dimension of $R$ is at most one. In case $R$ is a finite dimensional ...
1
vote
1answer
212 views

When does a subspace of the affine space form a regular sequence in a ring of regular functions?

Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$. Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{...