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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4 votes
1 answer
187 views

Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?

All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...
1 vote
1 answer
434 views

Non-linear Lie algebra

Several versions of non-linear Lie algebras exist - at least in the physics literature. One version is just an ordinary Lie algebra but where the underlying vector space is a polynomial algebra. ...
3 votes
1 answer
151 views

Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational

Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
1 vote
1 answer
165 views

Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
1 vote
1 answer
216 views

Wedderburn theorem for finite-dimensional algebras over the complex numbers

I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
3 votes
1 answer
193 views

Tangent space of a GIT quotient of $GL_{N}$

Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
5 votes
0 answers
178 views

From group ring to ring ring?

For a group $G$, the set $\mathbb{Z}[G]$ of all formal $\mathbb{Z}$ linear combinations is a ring with unit. Now the set $\mathbb{Z}[\mathbb{Z}[G]]$ gets the structure of a ring from the addition in $\...
3 votes
2 answers
298 views

Chirality of octonion algebras

Octonion multiplication can be defined with respect to a set of triads. A set of such triads can be represented by a directed Fano plane diagram such as the following two diagrams. This depicts two ...
2 votes
1 answer
626 views

A complete lattice of functions

Let $D$ be a set, $\mathbb{N_0}$ the set of natural numbers including zero. Let $P$ be the set of all functions from $D$ to $\mathbb{N_0}$, i.e. $P = \lbrace m \mid m: D \rightarrow \mathbb{N_0} \...
0 votes
0 answers
221 views

Fundamental theorem of algebra for sedenions

The Eilenberg–Niven theorem generalizes the fundamental theorem of algebra for quaternionic polynomials,¹ and this theorem was further generalized to also encompass octonionic polynomials.² Does ...
1 vote
1 answer
227 views

Zeroes of elementary polynomials without involving closed-form solutions

Consider the following two polynomials, where $n$ is an integer: $$ p_n(x) = x^3-\frac1nx-\frac2n, \\ q_n(x) = x^2-\frac2n. $$ For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
14 votes
2 answers
1k views

Is every group the automorphism group of a ring?

I know not all groups can be realized as the automorphism group of a group. For example, it is well-known that no group can have $\mathbb Z/n\mathbb Z$, with $n > 1$ odd, as automorphism group. Now ...
9 votes
1 answer
217 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
1 vote
0 answers
106 views

Is there a "natural" interpretation of the power function for octonions and for sedenions?

This question is a sequel to Is there a definition of $\log(x)$ for quaternion/octonion $x$? Since $\log(x)$ is multivalued even for complex $x \in \mathbb{C}$, it is impossible to define $\log(x)$ ...
3 votes
1 answer
140 views

Kernels and cokernels in a quotient of an abelian category

I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
2 votes
2 answers
163 views

Minimal ideals and subalgebras of semisimple algebras

I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...
1 vote
1 answer
282 views

A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$. Recursive definition of addition: $$x \oplus y := ((x+y) \...
5 votes
1 answer
237 views

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 ...
8 votes
1 answer
744 views

A "concrete" example of a one-sided Hopf algebra

I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition. To be more ...
4 votes
0 answers
292 views

Are there infinitely many simple integral fusion rings of rank $4$?

$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
-1 votes
1 answer
211 views

Can we classify all commutative unital algebras over the reals that are closed under $\sqrt{}$?

Can we classify all finite dimensional commutative (but not necessarily associative) unital algebras over the reals in which every element is a square?
1 vote
0 answers
56 views

Matroid for Laurent series

I am trying to find a matroid for profinite rings which are the inverse limit of their finite quotients, and whose linearly independent elements are of the form $L((t_1,\dots,t_n))$. To set this up, ...
1 vote
0 answers
45 views

generating set of polynomial ring

I am considering the polynomials $P=P[x_1,x_2,\ldots,x_n]$ with coefficients in a ring $R$. Consider a subset $S=\{p_1,p_2,\ldots,p_k\}$ of $P$. There is a map $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...
28 votes
6 answers
4k views

Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$. I ...
1 vote
0 answers
62 views

Groups with prescribed Ulm invariants

In Kaplansky's book infinite abelian groups he provides (through some exercises) a complete classification of $p^{\infty}$-torsion countable abelian groups in terms of Ulm invariants. In other words ...
0 votes
1 answer
60 views

Left quasi-inverse elements: motivation

An element $a$ in a ring $R$ is a left quasi-inverse if there exists $b\in R$ such that $a+b=ba.$ What is the motivation behind this definition?
3 votes
0 answers
160 views

Amalgamation of commutative subrings

Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$. Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection? This is equivalent to ask if in the ...
3 votes
1 answer
231 views

Motivational distinctions between max and min conventions in tropical geometry

I am aware that algebraically, there is no real distinction between the tropical semirings $A = (\mathbb{R} \cup \{ \infty \}, \text{min}, \infty, +, 0)$ $B = (\mathbb{R} \cup \{ - \infty \}, \text{...
24 votes
6 answers
2k views

What does the semiring of ideals of a ring R tell us about R?

Here is something I've wondered about since I was an undergraduate. Let $R$ be a ring (commutative, let's say, although the generalization to noncommutative rings is obvious). Ideals of $R$ can be ...
0 votes
0 answers
40 views

Nonassociativity in Cayley-Algebras

Let $(E,s)$ be a Cayley algebra over a unital commutative ring $A$ with unit element $e$ and $s$ an antiautomorphism (i.e. $s(uv) = s(v)s(u)$, $u,v \in E$) of $E$ such that $u + s(u) \in Ae$ and $N(u) ...
3 votes
0 answers
105 views

On the conditions for Artin-Schelter Gorenstein algebras

Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative). The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
1 vote
2 answers
441 views

Extension of the radical and radical of the extension of an ideal

If $A$ is a commutative ring, $I \subset A$ an ideal and $f:A \rightarrow B$ a ring homomorphism, then the extension of $I$, $I^e = \langle f(a): a \in I \rangle$ does not commute with the radical, I ...
6 votes
1 answer
157 views

Constructing countable threelds of finite dimension

A threeld is a generalization of a field, with three operations, such that the $F$ is a field with respect to the first (outer) and second (middle) operations (call it the outer field), and $F\...
13 votes
3 answers
1k views

Are the trace relations among matrices generated by cyclic permutations?

Let $X_1,\dots,X_n$ be non commutative variables such that $\operatorname{tr} f(X_1,\dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r \geq 1$. Does this ...
6 votes
0 answers
252 views

Lie algebra cohomology of the space of vector fields

For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1(...
2 votes
1 answer
72 views

Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?

The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
9 votes
1 answer
407 views

Towards the complex unit conjecture

After Gardam had found a counterexample to the Kaplansky unit conjecture for the group ring $K[G]$ with $K = \mathbf F_2$ and $G = P$, the Promislow group, Murray extended this to $\mathbf{F}_p[P]$ ...
61 votes
3 answers
7k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
77 votes
9 answers
6k views

Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?

The question is the extent to which we can unify addition and multiplication, realizing them as terms in a single underlying binary operation. I have a number of questions. Is there a binary ...
1 vote
1 answer
135 views

Cofactor an geometrical mean in $\mathit{SPD}_3$: a Gårding-like inequality

The cofactor map $A\mapsto\widehat A$ is polynomial homogeneous of degree $n-1$ over $\mathbf M_n({\mathbb R})$. It can be polarized into an $(n-1)$-linear symmetric map. When $n=3$, this provides a ...
3 votes
1 answer
219 views

Rings or algebras with many nilpotent elements and efficient computation

Crossposted from quantum.SE where comment appears to suggest that solving modulo 2 might be possible. Searching the web for '"quantum computer" nilpotent' returns many results, so maybe the ...
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 ...
4 votes
1 answer
341 views

How to have MAGMA work with subgroup of ATLASGroups?

I'm trying to work with various maximal subgroups of the Thompson sporadic group. The command Group("Th"); which works for some of the sporadic groups, ...
6 votes
0 answers
184 views

The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

I asked this question over a year ago on Math.StackExchange but I didn't get an answer. In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
3 votes
2 answers
213 views

Adjunctions and inverse limits of derived categories

Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its ...
3 votes
0 answers
194 views

Coevaluation for linear categories

For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...
1 vote
0 answers
212 views

Is the span of all nilpotent ideals also a nilpotent ideal?

Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...
5 votes
2 answers
394 views

How is the classification of groups extensions by $H^2$ related to Yoneda Ext?

It is well-known that group extensions $$1\to A \to H \to G \to 1$$ where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
3 votes
0 answers
128 views

Generalized wreath products of commutative algebras with Hopf algebras

Fix $k$ a commutative ring (or, if more convenient, assume it’s a field or even an algebraically closed field of characteristic 0, which is the case I’m mainly interested in). Let $A$ be a unital ...
5 votes
0 answers
276 views

Connections in non-commutative geometry

Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...

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