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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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 ...
Janik's user avatar
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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 ...
Mare's user avatar
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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 $J(A)$ ...
DiegoS10's user avatar
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 ...
user304582's user avatar
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}...
Fred Rohrer's user avatar
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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 ...
Mahsa Pavenejad's user avatar
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 ...
Lukas Woike's user avatar
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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 ...
R. Shhaied's user avatar
6 votes
0 answers
125 views

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 $\...
Tyler Lawson's user avatar
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14 votes
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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. ...
Benjamin Steinberg's user avatar
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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 ...
Fernando Muro's user avatar
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 ...
ABB's user avatar
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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 ...
Jie Pan's user avatar
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1 vote
1 answer
2k views

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 $...
toblatp's user avatar
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21 votes
2 answers
1k views

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 $w_i$ ...
pglpm's user avatar
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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 ...
katana_0's user avatar
  • 353
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?
user127555's user avatar
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 ...
ABB's user avatar
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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) = ...
Cornelius Brand's user avatar
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 ...
ABB's user avatar
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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 $B$ ...
ABB's user avatar
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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 ...
Drew Armstrong's user avatar
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)...
user521337's user avatar
  • 1,189
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 ...
Mare's user avatar
  • 25.8k
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 ...
André Henriques's user avatar
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 $A_{\mathbb{Z}}$ (...
solver6's user avatar
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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 $y^2-y\in K$, ...
E.R's user avatar
  • 31
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", ...
user44191's user avatar
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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 ...
Ilma Azzahra's user avatar
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 ...
Edwin Beggs's user avatar
  • 1,213
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 ...
Matt Ollis's user avatar
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 *_{\...
Henri Johnston's user avatar
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}...
Ben Webster's user avatar
  • 43.9k
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)...
ABB's user avatar
  • 3,898
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 ...
Henri Johnston's user avatar
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 ...
Simon Lentner's user avatar
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 ...
Denis T's user avatar
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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 $...
ABB's user avatar
  • 3,898
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 ...
user15749's user avatar
  • 111
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 ...
Mare's user avatar
  • 25.8k
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 ...
Sean Eberhard's user avatar
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 ...
Robert Bruner's user avatar
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 ...
C. Kwong's user avatar
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 $A$-...
Iteraf's user avatar
  • 482
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) ...
Anonymous Coward's user avatar
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?
Sara.T's user avatar
  • 151
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 ...
Sarah's user avatar
  • 39
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: ...
rschwieb's user avatar
  • 1,593
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, ...
Anupam Singh's user avatar
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 ...
Artor Waxsess's user avatar

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