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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
16 views

About extending maps to semi derivations

Let consider a map $d: X \to P$, where $X$ is a generating set of $P$ and $(P,\cdot, \{,\})$ is a free Poisson algebra. How can we extend $d$ to a semi derivation map from a subalgebra $P$ to $P$? By ...
1 vote
1 answer
106 views

Invariant ring of the subvariety

Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...
  • 533
5 votes
0 answers
174 views

Which properties of the pullback/restriction functor imply surjectivity of a ring homomorphism

Let $f:R\to S$ be a homomorhpism of (noncommutative) algebras over some field k (say the complex numbers) and let $F:Rep(S)\to Rep(R)$ be the corresponding functor between the categories of finite-...
-1 votes
0 answers
65 views

Show that $\beta$ is algebraic over F($\alpha$) [closed]

Question: Let E be an field extension of F, and let $\alpha,\beta \in E$. Suppose $\alpha$ is transcendental over F but algebraic over F($\beta$). My solution: Let F $\subseteq $ E, a,b $\in$E and ...
2 votes
1 answer
71 views

Proof of restrictableness of Lie algebra without basis

$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the ...
55 votes
23 answers
5k views

Results from abstract algebra which look wrong (but are true)

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
3 votes
0 answers
134 views

Definitions of torch ring

Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions. An FGC ring is a commutative ring whose ...
  • 1,448
-2 votes
0 answers
231 views

Algebraic geometry on rings by replacing "zero" by "zero divisor"

Let $R$ be a unital commutative ring. One can generalize the concept of algebraic set or variete in the following way: Let $T\subset R[x_1,x_2,\ldots,x_n]$ be a set of polynomials. We define $$Z(T)=...
2 votes
1 answer
55 views

How to define inverse of a non-degenerate triangular structure?

Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \...
2 votes
0 answers
109 views

Classifying indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$

I'm now interested in classifying the indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ : the group ring of $\mathbb{Z}/p\mathbb{Z}$ over the ring $\mathbb{Z}/p^{2}\...
  • 981
0 votes
2 answers
297 views

Torsion of modules

Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
  • 2,284
3 votes
1 answer
332 views

What is the name for algebras generated by elements, all of whose cubes vanish?

Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
  • 300
7 votes
2 answers
532 views

Deriving consequences of identities

Suppose we are given a variety in the universal algebra sense. For concreteness, suppose that we have two binary operations $+,\cdot$, three unary operations $-,\ast,'$, and two zeroary operations $0,...
  • 16.3k
1 vote
2 answers
294 views

Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
1 vote
0 answers
109 views

Structure theorem for finitely generated profinite abelian groups

Is there a structure theorem for finitely generated profinite abelian group like a structure theorem of f.g. abelian group?
  • 599
3 votes
0 answers
116 views

Summing over roots of a simple Lie algebra and Deligne series

For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...
  • 857
4 votes
0 answers
186 views

Road map for learning cluster algebras

I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
  • 533
5 votes
1 answer
209 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 ...
  • 19.7k
2 votes
0 answers
102 views

Construction of a certain long exact sequence

Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field. Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
  • 22.9k
0 votes
2 answers
120 views

Quotients of Koszul algebras

Let $A$ be a noncommutative Koszul algebra (see here for a definition of Koszul) and let $c \in A$ be a central element. Will the quotient of $A$ by the ideal generated by $c$ again be Koszul. If not ...
8 votes
0 answers
112 views

Behavior of $K_0$ towers

In the spirit of the automorphism tower problem, I've been thinking about "$K_0$ towers." Since $K_0$ may be imbued with a ring structure by the tensor product, it makes sense to ask what ...
  • 385
0 votes
0 answers
68 views

What are the properties of square-matrix algebra with this equivalence class?

Consider the set of all square matrices with the following equivalence class: $\mathbf{A}\sim\mathbf{A}\otimes\mathbf{I}_n$ (or, alternatively, as user @M.G. proposed, $\mathbf{A}\sim\mathbf{I}_n\...
  • 8,682
6 votes
0 answers
113 views

Rings with epimorphism from a finitely generated ring

For a commutative ring with unit $R$ let's say it has property $(*)$ if there is an epimorphism in the category of rings ${\mathbb Z}[X_1,\dots,X_n]\to R$, where the former is the polynomial ring in $...
  • 630
0 votes
0 answers
131 views

completion and tensor product

Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers? If $A$ is noetherian, it is clear because one has for $k$ a residue ...
  • 3,223
4 votes
1 answer
74 views

Real forms of the general linear Lie superalgebra

I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$. The real forms of the simple complex Lie superalgebras were classified by ...
0 votes
0 answers
45 views

Let $g$ be regular in $A$, when do we have $(gx_i-f_i)_{A[\underline{x}]}=A[\underline{x}]\cap (gx_i-f_i)_{A_g[\underline{x}]}$

Also asked in stackexchange. Let $A$ be a commutative unital ring. Let $g$ be a regular element of $A$. Let $A_g$ be the localization of $A$ at the set $\{g^n:n\geq 0\}$, then we have an injection $A\...
  • 299
1 vote
0 answers
85 views

When do two path algebras share an underlying graph?

Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction. Since ...
  • 385
3 votes
1 answer
249 views

Positive system of algebraic integers

Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
4 votes
2 answers
237 views

Cubical vs. simplicial Hochschild cohomology

Simplicial Hochschild cohomology. $\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\B}{\mathrm{B}}\newcommand{\Obj}{\mathrm{Obj}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Mod}{\mathsf{Mod}}$One way to ...
  • 1,143
7 votes
1 answer
554 views

Name for vector spaces with two algebra structures that satisfy the exchange law

Is there a name/reference for the following object? We have a vector space $V$ over some field with two associative bilinear operations $\circ,*:V \times V \to V$ which satisfy the interchange law, i....
1 vote
0 answers
76 views

Generating set of a group with a unique minimal normal subgroup

I started reading this paper by Andrea Lucchini. Title: Generators for Finite Groups with a Unique Minimal Normal Subgroup. Theorem 1.1(Main Theorem) If $G$ is a non cyclic finite group with a unique ...
  • 11
2 votes
1 answer
211 views

Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra?

Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{...
2 votes
0 answers
57 views

Example of a secondary representation of a module that is not a direct sum

Let $A$ be a commutative ring. An $A$-module $M$ is said to be secondary if $M\neq 0$ and for each $a\in A $, the endomorphism $\phi_a:M\to M$ defined by $\phi_a(m)=am$ for $m\in M$ is either ...
  • 621
-3 votes
1 answer
119 views

Exponential order of unipotent elements in an endomorphism ring of abelian groups

$\DeclareMathOperator\End{End}\newcommand{\Id}{\mathrm{Id}}$Let $E=\End(I)$ be the endomorphism ring of the abelian group $I$. We have the following statement for $B\in E$, $p$ a prime number and $r$ ...
  • 95
1 vote
0 answers
44 views

Existence of a minimal ideal with a specific property

Suppose that $R$ is a super-commutative ring (i.e. it is a unital $\mathbb{Z}_2$-graded ring satisfying $xy=(-1)^{|x|\cdot |y|}yx$ where $|x|$ denotes the grading degree of a homogeneous element $x\in ...
2 votes
0 answers
83 views

Question concerning relationships between different $p$-modular systems and Brauer character values

Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
3 votes
2 answers
313 views

What is an example of a Frobenius algebra that is not Koszul?

What is an example of a Frobenius algebra that is not Koszul? Are there reasonable requirements for a Frobenius to be Koszul?
2 votes
0 answers
73 views

Which definitions of "local module" have gotten traction?

It seems like "local module" has been defined a lot of ways: if 𝑀 has a largest proper submodule. (This math.se post) if it is hollow and has a unique maximal submodule (Singh, Surjeet, ...
  • 1,448
2 votes
0 answers
74 views

Is the associated algebra of a quadratic filtered algebra again quadratic

Let $A$ be a filtered quadratic algebra. Let $G(A)$ be the associated graded algebra. Will $G(A)$ again be a quadratic algebra or can higher relations appear when passing to the graded seeting? EDIT: ...
1 vote
1 answer
83 views

Two (or less) filtrations on coenveloping coalgebra

Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) ...
  • 3,712
2 votes
1 answer
83 views

Coproduct for a Frobenius algebra

The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
3 votes
1 answer
292 views

What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)

I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
3 votes
0 answers
166 views

Where could a paper on a unification of matrix decompositions be published?

I've got a paper which shows that when the spectral theorem (as a statement that every self-adjoint matrix can be unitarily diagonalised) is naively generalised to $*$-algebras other than the complex ...
  • 4,058
4 votes
0 answers
48 views

Real forms of complex Lie algebras with specified semisimple part

Let $\mathfrak g$ be a complex Lie algebra. If $\mathfrak g$ is semisimple then its real forms are completely understood. In particular there is a compact real form of $\mathfrak g$, which is unique ...
2 votes
0 answers
154 views

Ring homomorphisms from the commutative ring into $\mathbb{Z}_2$

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Spec{Spec}$Let $A$ be a commutative ring not necessarily with unit and $\mathbb{Z}_2 =\{0,1\}$ be the field of two elements. I am looking for a paper ...
7 votes
1 answer
161 views

Free median algebras and maximal linked systems

$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
  • 54.8k
1 vote
0 answers
63 views

Structure and representation of a non-homogeneous quadratic algebra

Let $m$ be a positive integer and $s$ be a fixed $m\times m$ matrix. Let $U$ be the unital associative algebra (over $\mathbb{C}$) generated by $\{u^i_{j}\}_{1\leq i,j\leq m}$ quotient over the ...
  • 271
4 votes
2 answers
539 views

A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?

A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
4 votes
1 answer
197 views

Wedderburn decomposition of special linear groups

$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
5 votes
0 answers
139 views

Bar constructions and pushouts

Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout. Is there any hope of ...
  • 1,440

1
2 3 4 5
59