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.
367 questions
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When does End(M) consist entirely of zero, zero divisors, and units?
Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see Rings in which every non-unit is a zero divisor for various stabs at what these are called). Let $M$ be ...
- 1,197
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Rigid monoidal and closed monoidal categories
I am trying to understand the relationship between rigid monoidal categories and closed monoidal
categories. First every rigid monoidal category is closed, with an adjoint to the functor $X \otimes -$ ...
- 1,144
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824
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Unitary unit conjecture for group rings
The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g....
- 151
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Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$
I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...
- 356
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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}...
- 2,356
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Maximal centralizer in full matrix ring
I will be so thankful if someone can help me with the following question.
Is it possible to obtain all maximal centralizers in the full matrix ring, $M_n(F)$, for an arbitrary finite field $F$? Here, ...
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Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
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Are quotients of polynomial rings almost UFDs?
If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\...
- 1,412
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
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Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
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Left-Module Structure on the Tensor Product ofTwo Left Modules
Given a noncommutative ring $R$, and two (left) $R$-modules $M$ and $N$, how does one define a left action on the the vector space tensor product $M \otimes N$? Multiplying on just the first factor of ...
- 389
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Unital *-homomorphisms between matrices
It is mentioned on Wikipedia that every unital *-homomorphism $\Phi:M_i\to M_j$ is necessarily of the form $\Phi(a)=U^*(a\otimes I_r)U$ for some unitary $U$ and some $r$. (Here $M_i$ are the $i\times ...
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Non-commutative rings where every non-unit is contained in a completely prime ideal
Below, all rings are associative and unital; and the word "ideal" always refers to a two-sided ideal.
Let's stipulate that a ring $R$ has property (P) if every non-unit of $R$ is contained ...
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Shape of axioms in algebraic structures
When defining algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in many ...
- 1,341
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$\text{Mod}(A)$ is an $E_n$ category $\Leftrightarrow$ $A$ is an ??? algebra
Say we're working in a symmetric monoidal $\infty$-category $\mathcal{S}$, and $A$ is an associative algebra in it. For instance,
$$\mathcal{S}\ =\ \text{dg vector spaces},\ \ \ A\ =\ \text{a dg ...
- 5,701
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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 $mz=z=zm$ ...
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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 ...
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Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...
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Orders of Clifford algebra
Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.
Q1) Is it ...
- 1,692
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Existence of non-trivial reflexive modules
Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f_M:M \rightarrow M^{**}$ (where $M^{*}=Hom_R(M,R)$) is an isomorphism, where $f_M(m)=g$ with $g(h)=h(m)$, ...
- 26.5k
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Are PD-, $\lambda$-, $\psi$-, and $\delta$-rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
- 11.8k
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Cancellable elements of a power semigroup
For a semigroup $S,$ its power semigroup $P(S)$ is the semigroup of all non-empty subsets of $S$ with the operation given by $AB=\{ab\,|\,a\in A,b\in B\}.$ I would like to know about the cancellable ...
- 1,445
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348
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Do Frobenius subalgebras form a lattice?
A finite-dimensional, unital, associative algebra $A$ over a field $k$ is termed a Frobenius algebra if it is endowed with a nondegenerate bilinear form $\sigma : A \times A \to k$ satisfying the ...
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Name for a Hopf algebra whose only grouplike element is the identity?
For a $k$-Hopf algebra $H$ and element $h \in H$ is called grouplike is $\Delta(h) = h \otimes h$ and $\epsilon(h)=1_k$ ($\epsilon$ is the counit). The identity $1_H$ is clearly grouplike, but in ...
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Subalgebras of singular matrices (less naive version)
Is it true that, for any subalgebra $\cal S$
of the algebra of linear operators in a finite-dimensional vector space $V$ over a field,
$$
\bigcap_{A\in\cal S}\ker A=\{0\}\hbox{ and }
\bigcup_{A\in\cal ...
- 3,932
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Non-associative commutative "group"
When dealing with some hash functions that I was trying to speed up, I toyed with a binary operation with the goal to "approximate" the addition on $\{0,1\}^*$ when seen as binary ...
- 48.9k
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Rank of sum of Galois conjugates of a matrix
Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in \mathrm{...
- 1,810
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Prime/irreducible elements in certain (integral) ring extensions
The answer to this question says the following:
Let $R$ be a finitely generated $k$-algebra, where $k$ is a field.
If $p \in R$ is a prime element, then $p$ is a prime element in $\tilde{R}$, the ...
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On skew monoid rings and skew ordered series rings
To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
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What is the relationship between the Dirac algebra and the Clifford algebra?
While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is ...
- 3,515
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Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
- 10.5k
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Splitting the Witt vectors of $\overline{\mathbb{F}_p}$
Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\...
- 2,052
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Commutative associative rational binary operations
What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...
- 19.7k
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A hands-on description of a "completion" of the free commutative monoid on countably many generators
This is basically an I'm-weak-at-algebraic-geometry question. I asked it as a warm-up question here, but Ilya N asked me to break that post up into several questions.
Consider the free commutative ...
- 54.6k
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Comparing different Euclidean algorithms on a Euclidean domain
I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
- 2,454
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Nilpotent operator of the Weyl algebra
For a research project I'm currently working on, I came across the following problem:
Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where $...
- 499
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chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,...,x_n,...]$
Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?
Question: Is there any maximal ...
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How does Constructive Quantum Field Theory work?
Please correct me if I'm wrong, but it seems to me that two and three dimensional axiomatic quantum field theory were constructed as follow: the wightman axioms were formulated in euclidean space via ...
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Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?
Motivated by this question, it seems natural to ask the following:
Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
- 4,432
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Is the universal enveloping algebra of a finite-dimensional Lie algebra (left) noetherian?
The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is a flat deformation of $S(\mathfrak{g})$, so these algebras should be similar in many ways. Does at least this general similarity ...
- 53
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447
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About a categorical definition of graded (coloured) algebra
The definition of graded algebra had a growing interest in algebra and mathematical physics (see $[GTC]$), I see that this topic has an elegant and simple categorical generalization, but I have not ...
- 4,165
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655
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Are all algebras Igusa-Todorov?
A finite dimensional algebra A is called (n-)Igusa-Todorov in case there exists a module V such that for any module M there is an exact sequence:
$0 \rightarrow V_2 \rightarrow V_1 \rightarrow \Omega^{...
- 26.5k
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Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
- 356
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Linear maps preserved by algebra automorphisms
Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field.
Let $s_A:A\to F$ be an $F$-linear map.
Now consider an arbitrary field extension $K/F$, and define $s_{A\...
- 1,766
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385
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Which monoids can be realized as the monoid of ideals of a commutative monoid?
Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an ideal of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) ...
- 10.5k
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When are all modules direct factors of a direct product of a fixed one?
Background: For a ring $R$, we denote by ${\rm\mathop Mod}(R)$ the category of all (say right) $R$-modules. If $R$ is pure semisimple, then it is known that ${\rm\mathop Mod}(R)={\rm\mathop Add}(M)$, ...
- 666
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210
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A presentation for a subalgebra
Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$.
Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...
- 5,039
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Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?
Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$.
The $n$-th type-A subdivision ...
- 33.8k
3
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1
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158
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Local Frobenius algebras and their opposite algebras
Assume all algebras are finite dimensional quiver algebras over a field (no restriction of generaltiy if the field is algebraically closed).
Let A be a local Frobenius algebra.
Is A isomorphic to its ...
- 26.5k
3
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1
answer
328
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What does it mean for the surreal numbers/partizan games to be "universally embedding"?
In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
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