ra.rings-and-algebras's questions - English 1answer

1.928 ra.rings-and-algebras questions.

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 ...

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 ...

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{...

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 ...

Let us say that a mathematical structure of cardinality $\omega_1$ is Jonsson whenever every one of its proper substructures is countable. There are examples of Jonsson groups due to Shelah or ...

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 ...

A commutative Bezout ring with $1$ is a ring in which every finitely generated ideal is principal. Is there any charactrization for indecomposable semi local Bezout ring? For example , for a ...

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 ...

I m teaching linear algebra and encounter this theorem: two matrices A and B are similar iff tI - A and tI - B are equivalent (as polynomial matrices), where I is the unit matrix. The proof that I ...

Let $A,B$ be complete Boolean algebras and $\varphi,\psi:A\rightarrow B$ be maps preserving $0,1$, and arbitrary joins and meets. Let $C$ be the equalizer of these two; so $C=\left\{a\in A:\varphi(a)=...

Let $S$ be a finite set. Let $R$ be a complex vector space with basis indexed by subsets of $S$. Define a product on $R$ by defining it on the basis elements as $1_A\cdot 1_B=1_{A\Delta B}$, where $A\...

Let $A$ be a fixed boolean ring. Is there a sort of classification of boolean rings $B$ with $A \subseteq B$? For example, if $A=\mathbb{F}_2$, the answer would be Stone duality: $B=C(Spec(B),\mathbb{...

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 ...

Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread. Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...

I am studying theorem (1.11)* from this article: https://sci-hub.tw/https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-9.2.337 Theorem (1.11)*. Let $R$ be a Jacobson ring, then $S = R[x,\alpha]$ ...

Which ring is the maximal subring of matrix ring $M_{n}\left(R\right)$.n is the given number,maximal subring means maximal dimension.

I am interested in algebras whose subalgebra lattice is supremum-founded. Let us call those algebras small. A complete lattice $(L, \leq)$ is called supremum-founded, if for any two elements $x < ...

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 $\mathcal{C}_2=B\operatorname{-mod}$, ...

Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard ...

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}^...

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}...

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+...

Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated? Basically I want to believe I can ...

Let $R$ be a commutative ring with $1$. For an ideal $I$ of $R$ the $J$-radical of $I$, denoted by $J-rad(I)$, is the intersection of all maximal ideals of $R$ containing $I$, that is, $J-rad(I)=\cap_{...

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 ...

Is there known structures/ or has there been studies on $Hom(L_1,L_2)$ of distributive lattices? Could it be made into a lattice naturally? Is there any structure on the set of ring valued functions $...

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*?

I believe that the 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 am sure this is ...

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 $\...

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 $Y_1, \dots, Y_n$ ...

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{...

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 ...

Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...

I'm looking for some function $P: M \rightarrow R$, where $M$ is matrix ring over some non-commutative ring that would behave "like" rank. In particular: $P(A+B) \leq P(A)+P(B)$ $P(A\times B) \leq P(...

In the book “Algebraic Number Theory” written by Cassels and Frohlich, module index is defined. Let R be Dedekind domain, K be its quotient field, U be a n-dimensional vector space over K, and L,M be ...

Can the usual definition of a Lie algebra via commutators be simply adapted to quantum Lie algebras? Graphically you have the IHX scheme, with the X being a virtual crossing (so to say). Does it ...

In the theory of finite dimensional algebras there are many homological conjectures. When working over an algebraically closed field it is well known that any such algebra is Morita equivalent to a ...

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...

I’m now thinking about the property of category of semirings Rig. Is it complete or co-complete? I think that Rig has projective and inductive limits, and finite products and co-products, so it ...

Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to ...

Let $A$ be an arbitrary (not necessarily finite-dimensional) associative algebra over an algebraically closed field $K$, and let $\mathrm{fin\,}A$ denote the category of finite-dimensional $A$-modules....

An algebraic group is a group that is also an algebraic variety. There is also a theory of algebraic monoids. Is there are version of universal algebra that incorporates these examples, and other ...

I was wondering if anyone happens to know results for diagonalizing matrices over semirings. I was able to find a result for commutative rings: https://www.sciencedirect.com/science/article/pii/...

For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we ...

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...

By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting. It is easy ...

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 ...

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. ...

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