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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Canonical form for a pair of quadratic forms

Could anyone recommend a reference to a canonical form for a pair of quadratic forms over R (not necessarily positively definite)? This is probably related to the Weierstrass elementary divisors (but ...
mikhail skopenkov's user avatar
5 votes
2 answers
484 views

Sub-Hopf algebras of group algebras

Let $k$ be a field and $G$ a finite group. Is every sub-Hopf algebra over $k$ of the group algebra $k[G]$ of the form $k[U]$ for a subgroup $U$ of $G$ ?
Ralph's user avatar
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Pairwise orthogonality for partitions of unity in a *-algebra

Let $\mathcal{A}$ be a $*$-algebra with unit $1_{\mathcal{A}}$. As in the $\mathrm{C}^*$-setting, a projection is an element $p\in\mathcal{A}$ such that $p=p^2=p^*$. A partition of unity is a finite ...
JP McCarthy's user avatar
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1 answer
340 views

A question on linear algebra over non-Archimedean local field

Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it ...
asv's user avatar
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Injective modules

Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。
Sun YongLiang's user avatar
5 votes
3 answers
555 views

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
ghc1997's user avatar
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Ring of continuous functions is a Jacobson ring

Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
Serge the Toaster's user avatar
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Are lacunary functions still lacunary over rings larger than $\mathbb{C}$?

Take a lacunary function of your choice ex: $$ f(z) = z + z^2 + z^4 + \cdots = \sum_{k=0}^\infty z^{2^n} $$ Obviously this cannot really be analytically or meromorphically continued outside the unit ...
Sidharth Ghoshal's user avatar
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1 answer
300 views

What is the smallest rank for a noncommutative fusion ring?

A fusion ring $\mathcal{F}$ (of rank $r$) is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying ...
Sebastien Palcoux's user avatar
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
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2 answers
386 views

Lefschetz Principle for semisimplicity

I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like. Let $R$ be a unital ring (not necessarily ...
Benjamin Steinberg's user avatar
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1 answer
405 views

Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms $f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$ for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put $R^{\...
Pierre's user avatar
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1 answer
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Two-sided bar construction

On page 4 of this paper by H. Abbaspour, the author defines the two-sided bar construction $$B(A,A,A):=A\otimes T(s\bar{A})\otimes A$$ of a differential graded algebra $(A,d_A)$ (over a field). The ...
Dave's user avatar
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3 answers
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Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let $\...
Leo's user avatar
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2 answers
472 views

Lie algebra embeddings and the center of their enveloping algrabras

Let $\mathfrak{g}_1\subset \mathfrak{g}_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset \mathfrak{sl}(...
Alex's user avatar
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1 answer
250 views

Well defined Tensoring of spectral triples

Hi, I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it. Question: In connes standard model he takes ...
SMF's user avatar
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3 answers
668 views

Does every compact Hausdorff ring admit a decomposition into primitive idempotents?

Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...
Niemi's user avatar
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1 answer
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Examples and importance of Embedding (and Non-Embedding) Theorems

An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
5 votes
1 answer
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Can the Knaster-Tarski theorem be proved using the Schroeder-Bernstein theorem?

The reverse can be done easily and the proof is well known I am wondering if the exact same argument can be used to prove reverse as well.
user10122's user avatar
5 votes
1 answer
546 views

What is an exponential?

Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential ...
Steve Huntsman's user avatar
5 votes
1 answer
175 views

Are module finite algebras over semiperfect rings again semiperfect?

Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
uno's user avatar
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1 answer
201 views

Equivalences of categories of complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$. Let $R, S$ be two commutative $k$-algebras. Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
Walterfield's user avatar
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2 answers
332 views

Writing $1-xyzw$ as a sum of squares

Can you write $1 - xyzw$ in the form $p + q (1 - x^{2}-y^{2}-z^{2}-w^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z,w]$? For ...
Abhiroop Sanyal's user avatar
5 votes
1 answer
187 views

Subalgebras of singular matrices

Is it true that any subalgebra of singular matrices have a common null-vector? In other words, is it true that, for any subalgebra $\cal S$ of the algebra of linear operators in a finite-dimensional ...
Anton Klyachko's user avatar
5 votes
1 answer
94 views

Tensor decomposition under derived equivalence

Let $K$ be a field. Suppose $A$ and $B$ are $K$-algebra and there is a derived equivalence $F:D^b(A)\cong D^b(B)$ between their bounded derived categories. If we assume that $A$ has a tensor ...
Master Gang's user avatar
5 votes
1 answer
119 views

Integral monoid rings and Ore conditions

Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$. I have two, ...
Simone Virili's user avatar
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1 answer
698 views

What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module?

Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E_2$ algebra $HH^{\bullet}...
davik's user avatar
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5 votes
2 answers
611 views

Characters on Hopf algebras

For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
Fofi Konstantopoulou's user avatar
5 votes
3 answers
2k views

Ideal structure of a tensor product of certain algebras

I would be grateful if anyone could give me a reference regarding the following question. Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...
Ilja's user avatar
  • 423
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1 answer
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$Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$

$G$ 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 ...
Cooler Panda's user avatar
5 votes
1 answer
1k views

Periods in the trivial extension algebra of the incidence algebra of the divisor lattice

Definition of $C_L$ for people who like number theory: Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the ...
Mare's user avatar
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5 votes
1 answer
220 views

Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum

Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A_{q_n}$ as in this paper. Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of ...
MathStudent's user avatar
5 votes
1 answer
173 views

an algebra generated by some known series

Denote the e.g.f. for the number of (unordered) rooted labeled trees on $n$ nodes by $$\Phi(x)=\sum_{n\geq1}\frac{n^{n-1}}{n!}x^n.$$ And, the related series $\Psi(x)=\sum_{n\geq1}\frac{n^n}{n!}x^n$. ...
T. Amdeberhan's user avatar
5 votes
1 answer
192 views

Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?
M92's user avatar
  • 357
5 votes
3 answers
245 views

Matrices over Finite Prime Fields

Let $p$ be an odd prime and $\mathbb Z_p$ be the prime field of order $p$. Consider the matrix ring $R=M_n(\mathbb Z_p)$. Is there any method to count the solutions of the equation (in the ring $R$) $...
zacarias's user avatar
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5 votes
1 answer
501 views

Self-injective basic algebras

Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective? The algebra $A$ cannot be finite-dimensional, ...
Fernando Muro's user avatar
5 votes
2 answers
1k views

Is there a notion of congruence relation for essentially algebraic structures?

In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations. A congruence ...
Peter Arndt's user avatar
5 votes
1 answer
158 views

Relation between row space and column space resp. null space and left null space over general rings

Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
Thomas Preu's user avatar
5 votes
1 answer
158 views

Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)? I'm under the impression that when ...
rschwieb's user avatar
  • 1,593
5 votes
1 answer
233 views

Categorical Morita equivalence implies equivalence of module categories?

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true ($R$-Mod) $\simeq$ ($S$-Mod). $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective ...
Student's user avatar
  • 5,008
5 votes
1 answer
109 views

Fredholm elements of a Lie algebra

An element $a$ of a Lie algebra $L$ is called a Fredholm element if the adjoint operator $\mathrm{ad}_a:L \to L$ is a Fredholm linear map. That is: its kernel is a finite-dimensional space and its ...
Ali Taghavi'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
5 votes
1 answer
494 views

Question about denoting/designating of algebraic structures

I saw this image on Wikipedia (Template:Group-like structures, current revision): Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, ...
user avatar
5 votes
1 answer
479 views

Extending an automorphism from a sub-algebra to the algebra

Let $A \subseteq B$ be two (associative with $1$) $k$-algebras, where $k$ is a field of characteristic zero, and let $f$ be a $k$-automorphism of $A$. I am interested to know 'when' one can extend $f$...
user237522's user avatar
  • 2,783
5 votes
1 answer
281 views

Tame-Wild Dichotomy theory for infinite dimensional Hereditary algebras

A famous theorem of Drozd says that every finite dimensional hereditary algebra is either of tame or wild representation type. I am interested in infinite dimensional hereditary algebras. Is there a ...
batconjurer's user avatar
5 votes
1 answer
336 views

Can the trivial module be stably free for a monoid ring?

Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ ...
Benjamin Steinberg's user avatar
5 votes
1 answer
484 views

When is the adjacency algebra of a graph an association scheme?

The adjacency algebra of a graph is the algebra consisting of all polynomials in the adjacency matrix of the graph. An association scheme is a commutative matrix algebra containing the identity and ...
David Roberson's user avatar
5 votes
1 answer
360 views

Classification of indecomposable modules in tame hereditary algebras

An algebra $A$ is said to be tame if the isomorphism classes of indecomposable $A$-modules in each dimension occur in a finite number of 1-parameter families. $A$ is said to be of finite ...
Iteraf's user avatar
  • 482
5 votes
1 answer
531 views

When is the category of Gorenstein projective $R$-modules Frobenius?

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a ...
Matthew Pressland's user avatar
5 votes
2 answers
335 views

Gelfand-Kirillov dimension of generalized Weyl algebras

I believe that the Gelfand-Kirillov (GK) dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$. Does anyone have a reference for this? I can find partial results, and I ...
No1729's user avatar
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