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.
3,343
questions
7
votes
0
answers
171
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Wedderburn decomposition of twisted group algebras
Let $K$ be a field and $G$ be a finite group. Maschke's theorem states that the group algebra $KG$ is semisimple iff $|G|$ is not divisible by $\text{char}K$. In particular, if $\text{char}K=0$, the ...
5
votes
2
answers
318
views
Questions on weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called weakly symmetric in case $soc(P)=top(P)$ for every indecomposable projective module $P$ and it is called symmetric in case $D(A) \cong A$ as ...
7
votes
1
answer
202
views
The Image of a Derivation is Contained in the Jacobson Radical
Let $A$ be a finite-dimensional unital commutative associative algebra over a field $K$ of characteristic $0$. Is it true that for any derivation $D$ of $A$ we have $D(A) \subseteq J(A)$ where $J(A)$ ...
1
vote
0
answers
343
views
Intersection condition for polynomial ring and maximal ideals
In ring theory, there is interest in a condition known as the intersection condition. There is a brief comment in McConnell-Robson along these lines: Consider the ring $R = k[x,y]$ where $k$ is a ...
2
votes
0
answers
60
views
Linearity of a canonical morphism related to scalar extension and coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...
3
votes
1
answer
94
views
What is K+M structure?
In the following paper (Example 2.1), it has been mentioned to K+M to provide an example of a pseudo valuation domain which is not a valuation ring, and its reference is Gilmer's book, but I have no ...
7
votes
0
answers
218
views
Hochschild-Mitchell Homology
There is the notion of Hochschild-Mitchell homology for a $k$-linear category $\mathcal{C}$ (here $k$ can be a field). The definition is straightforward and so are some general properties, but somehow ...
5
votes
0
answers
112
views
The cyclic modules and self injective ring
It is well-known that if R is a Noetherian ring, and every finitely generated right R-module embeds in projective, then R is a self-injective.
My question is that could one replace "finitely ...
6
votes
0
answers
125
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Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
14
votes
1
answer
558
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.
...
2
votes
0
answers
74
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Units in the (stable) center of a Frobenius algebra [duplicate]
Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...
1
vote
0
answers
40
views
Relation between left projections
Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$.
Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...
8
votes
1
answer
558
views
Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics
Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...
1
vote
1
answer
2k
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Lanczos algorithm for finding $k$ smallest eigenvector
I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
21
votes
2
answers
1k
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Structures of the space of neural networks
A neural network can be considered as a function
$$\mathbf{R}^m\to\mathbf{R}^n\quad
\text{by}\quad x\mapsto w_N\sigma(h_{N-1}+w_{N-1}\sigma(\dotso h_2+w_2\sigma(h_1+w_1 x)\dotso)),$$
where the $w_i$ ...
7
votes
2
answers
452
views
The soccer splitting problem in arbitrary commutative ring
There's a folklore problem:
Let $x_1, \cdots, x_{23} \in \mathbb{Z}$ be the weights of $23$ soccer players. Now Master Yoda want's to form two soccer teams with $11$ players each. Turns out for ...
0
votes
1
answer
601
views
Book on algebraic structures
What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?
1
vote
0
answers
37
views
The statue of a sequence of finite projections
Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false?
Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...
7
votes
1
answer
282
views
A commutative variant of the exterior algebra
Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = ...
1
vote
0
answers
38
views
something concerning finite projections
Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.
Q. Can we say that ...
1
vote
2
answers
125
views
Upper triangular $2\times2$-matrices over a Baer *-ring
Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices
$\left(\begin{array}{cc}
a_1& a_2 \\
0 & a_4
\end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...
9
votes
1
answer
346
views
Origin of the concept of "homomorphism"? [duplicate]
When was the concept of a "homomorphism" of algebraic structures first introduced?
Steinitz' 1910 paper Algebraic Theory of Fields is often pointed to as the first true work of abstract algebra, yet ...
1
vote
0
answers
112
views
A generalized Cauchy type functional equation
Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...
2
votes
1
answer
194
views
$Ext_A^1(J,J)$ for the Jacobson radical $J$ of an algebra $A$
Let algebras be finite dimensional over a field $K$ and let $J$ denote the Jacobson radical (this is the intersection of all maximal right ideals) of an algebra. Being hereditary means that the ...
11
votes
3
answers
315
views
Unbiased Hopf algebras
In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
1
vote
0
answers
140
views
Derivations of special rings
Consider $p$-adic field $\mathbb{Q}_p$ and let $A$ be any finite dimensional $\mathbb{Q}_p$ algebra without zero divisors (and maybe without $1$). Now we can look on $A$ as on a ring $A_{\mathbb{Z}}$ (...
3
votes
0
answers
186
views
Ideals and Idempotents in a commutative ring
Let $I$, $J$, and $K$ be pairwise comaximal ideals of a commutative ring $R$ with $1$ with the property that if $x^2-x\in I$, then there exists $e^2=e\in J$ such that $x-e\in I$ and if $y^2-y\in K$, ...
3
votes
1
answer
475
views
Relation between coefficients of expansions
Related to Relations between coefficients of expansions of a rational function at 0 and infinity
I commented at the linked question that the question seemed less about what happened "at infinity", ...
18
votes
3
answers
2k
views
Why should we study derivations of algebras?
Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...
2
votes
0
answers
46
views
generalisations of module maps
For an algebra $A$ over a field $\mathbb{K}$, consider left modules $E,F$. The two obvious collections of maps to take from $E$ to $F$ are the left module maps ${}_A\mathrm{Hom}(E,F)$ and the linear ...
4
votes
0
answers
211
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
4
votes
1
answer
826
views
Twisted group rings and cohomology
Let $R$ be a commutative ring with unity and let $G$ be a finite group. Let $\gamma \in Z^{2}(G,R^{\times})$ be a $2$-cocycle. The twisted group ring (of $G$ over $R$ with respect to $\gamma$) $R *_{\...
12
votes
1
answer
643
views
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}...
2
votes
1
answer
74
views
Strongly finite projections in $*$-rings
Let $A$ be a $*$-ring. Let us have some points:
i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.
ii) On the set of projections, we write $p\leq q$ if $pq=p$.
iii)...
4
votes
1
answer
208
views
Cancellation property for certain integral group rings
We say that the cancellation property holds for a ring $R$ if for finitely generated projective $R$-modules $P$ and $Q$ we have that $R \oplus P \cong R \oplus Q$ implies $P \cong Q$.
In the case ...
1
vote
1
answer
159
views
Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$
Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked.
I know that there ...
13
votes
2
answers
820
views
Example of a ring with non-finitely generated unit group?
The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...
2
votes
0
answers
68
views
Transmission of finite projections
Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$).
Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...
5
votes
1
answer
394
views
Counterexample for the Skolem-Noether Theorem
If a division ring is finite-dimensional over its center then we can apply Skolem-Noether theorem (which asserts that every endomorphism is a conjugation).
Can someone give a counterexample of the ...
7
votes
1
answer
237
views
Is being a Frobenius algebra a rare condition for local algebras?
Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the ...
7
votes
1
answer
394
views
Extension-field subgroups of $\operatorname{GL}(n, K)$
$\newcommand{\op}[1]{\operatorname{#1}}\def\GL{\op{GL}}$I am interested in the subgroups of $\GL(n,K)$ of the form $\GL(n/s, E)$ for $E$ some extension field of $K$ of degree $s$. These arise in the ...
9
votes
1
answer
668
views
Curious anti-commutative ring
Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
0
votes
1
answer
3k
views
Cholesky decomposition – non-positive definite matrix
In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
5
votes
1
answer
167
views
Given a representation-infinite algebra, when is every AR component infinite?
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
6
votes
1
answer
360
views
Is every (left) graded-Noetherian graded ring (left) Noetherian?
I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...
5
votes
0
answers
181
views
Which rings are the endomorphisms ring of some abelian groups?
Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
3
votes
1
answer
108
views
Extensions of modules of type $FP_n$
Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely ...
9
votes
0
answers
139
views
Where is it shown that a countable self-injective ring is semilocal?
In Lawrence, John. "A countable self-injective ring is quasi-Frobenius." Proceedings of the American Mathematical Society (1977): 217-220. the first line is this:
...
3
votes
1
answer
168
views
Hermitian forms over $K\times K$
Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$.
When $V$ is a free module, ...
5
votes
1
answer
277
views
When is a zero dimensional local ring a chain ring?
A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that a zero dimensional local ring is a chain ...