## New answers tagged ra.rings-and-algebras

1
vote

Accepted

### On reflexive bialgebras

There shouldn't be any relations in general. Given an algebra $A$, there can be multiple different co-multiplications on $A$ to make it a bialgebra, or even a Hopf algebra. For instance, let $M(G)$ be ...

0
votes

### What are hypergroups and hyperrings good for?

I will try to say something about hypergroups (more generally about the hyperstructures) and hopefully it be useful and can answer to the questions: How is a canonical hypergroup to be thought of as ...

0
votes

### Is a "separable" algebra over a field finite-dimensional?

That $A$ is finite-dimensional by condition 2 is proved in Algebra: Chapter VIII by Bourbaki, p.232 Theorem 1, where separable algebra in sense of condition 2 is called "absolutely semisimple ...

2
votes

### Why should we study derivations of algebras?

Here’s a practical reason to look at derivations, from one algebraist’s perspective, broken down in a few steps.
Suppose you would rather look at the group of automorphisms of the algebra, instead of ...

1
vote

### Equivalent definition of Spin group in terms of automorphisms

$
\newcommand\R{\mathbb R}
\newcommand\Cl{\mathrm{Cl}}
\newcommand\tr{\mathrm{tr}}
\newcommand\form[1]{\langle#1\rangle}
\newcommand\Orthog{\mathrm O}
\newcommand\Pin{\mathrm{Pin}}
\newcommand\Spin{\...

6
votes

Accepted

### Reference about cancellation property for semigroups

A semigroup with this property seems to be called a right reductive semigroup (Wikipedia, nLab).

2
votes

Accepted

### Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$

Not much can be said about $F$. The second conditions should hold for $\mathbb C(u,v)$ for $u$ and $v$ two "generic" algebraically independent polynomials.
For an explicit construction, take ...

4
votes

Accepted

### Relationship between units of a ring and primitive characters of the ring under addition

The rings $\mathbb{Z}/n$ are the only examples.
I assume that "primitive character" just means that it is faithful, or equivalently that it does not factor through a proper quotient; this is ...

0
votes

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

It seems that there cannot be a natural interpretation, because there is no canonical definition for the meaning of $x^y$.
There is already no canonical definition when $x$ is a complex number (that ...

1
vote

### A general form of a maximal totally isotropic subspace in the split octonion algebra

Yes to both questions!
Every maximal totally isotropic $V$ is of this form by Theorem 3 on page 164 of
van der Blij, F.; Springer, T. A., Octaves and triality, Nieuw Arch. Wiskd., III. Ser. 8, 158-169 ...

4
votes

### How fast does the number of "fixed" points grow compared to the size of the ball in the following group?

One can describe $M/N$ by using basic commutative algebra.
Thing of $M$ as a free module of rank 5 over the polynomial ring $\mathbf{Z}[t,t^{-1}]$, with basis $(e_j)_{1\le j\le 5}$. So what you denote ...

5
votes

### How fast does the number of "fixed" points grow compared to the size of the ball in the following group?

The subgroup $F$ (I have not checked this is indeed the set of fixed points of $\phi$) is isomorphic to $\mathbb Z$, the question is just whether it is distorted or not, and if it is how much?
I'll ...

2
votes

Accepted

### Infinite-dimensional, non-unital Frobenius algebras

It is studied under the name "nearly Frobenius algebras" in this paper. In Example 3.3, the algebra $\mathbb{C}[[x,x^{-1}]]$ of Laurent series can be endowed with (countably) many unital but ...

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