# Questions tagged [division-ring]

Use this tag for questions about division rings in abstract algebra and/or noncommutative algebra.

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### How can I create the presentation (min. set of relations) of the quaternion group of order 8? How can I look for conjugation relations?

What are the steps that leads me to know the presentation (specifically defining the generators and the minimum set of relations required to define this group)of the quaternion group of order 8? What ...
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### when does a quaternion algebra isomorphic to $M_2(F)$?

We also suppose that the characteristic of a field is not $2.$ Definition 1. An algebra $B$ over $F$ is a quaternion algebra if there exist $i,j\in B$ such that $1,i,j,ij$ is an $F$-basis for $B$ ...
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### An example of a ring which is very close to division ring but not a division ring.

Let $R$ be a ring with unity. An element $a\in R$ is said to be a unit element if there exists $b\in R$ such that $ab=ba=1$. The ring $R$ is called a division ring if every nonzero element is a unit ...
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### Desargues $\implies$ associativity: Projective planes over non-associative structures?

I've been reading about constructing projective planes over division rings (skewfields). There's this very nice fact that if Pappus's theorem holds in a division ring, this ring is actually ...
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### Some problems in Noncommutative Algebra book of Benson Farb and R. Keith Dennis [duplicate]

In this, page 48, Exercies in chapter 1, there is a following exercise. Exercise 2. Let $R$ be a ring (with $1$) such that the only left ideals of $R$ are $0$ and $R.$ Show that $R$ must be a division ...
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### An exercise about Division Algebra

In this, page 48, Exercies in chapter 1, there is a following exercise. Exercise 1. Let $D$ be a division algebra which has finite dimension over the field $k.$ For each $a\in D$ show there is a monic ...
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### For Desarguian projective planes, coordinatization is inverse to the $K \mapsto K\mathbf{P}^2$ construction

In J. C. Baez's February 9, 2000 entry on his website it is claimed that the process of coordinatizing a Desarguian projective plane (i.e. obtaining a skew field/division ring in the style of the ...
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### A question to Finite Multiplicative subgroups in a division ring of I. N. Herstein

In this, I can't find the results in German as proof steps of Lemma 3 (... by Satz 88 [2, p. 72]) and Theorem 7 in page 123 (... Using results about division subalgebras of division algebras [1, p. 42,...
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### Why are real numbers on the number line but complex numbers aren't?

This seems like a question that should be relatively easy to answer, but for the life of me I simply can't figure it out. My question is relatively simply put in the title, and the answer seems like ...
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### Ring of Right Fractions is Spanned by $\theta^k, 0\leq k\leq d-1$

Suppose $D$ is a division ring such that every subring of $D$ is a right Ore domain, that is, if $a,b\in D\setminus\{0\}$ then $aR\cap b(D\setminus\{0\})\neq\emptyset$. Let $R$ be a subring of $D$. ...
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### What is minimal polynomial over a skew field?

I am reading "Polynomial extensions of skew fields" by J. Treur (see here.) What does he mean by "minimal polynomial $p$ of $\theta$ over skew field $K$"? Can we define the minimal ...
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### Let V be an infinite dimensional vector space over a division ring D. The Set F={θ:V→V: Im(θ) is a finite dimensional subspace of V} is a simple ring.

I have been able to prove that F is a proper ideal of End(V) but however stuck to show that F is a simple ring. My idea is to start with a two sided ideal of F, I assuming I≠0, then using a nonzero ...
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### Division Ring and Integral Domain

In abstract algebra, there are division ring, integral domain and field. Division Ring is a Ring when all elements are unit. Integral Domain is a Ring with no div '0' and commutative. Is there any ...
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### Show that a finite domain is a division ring [duplicate]

Let $R$ be a finite ring. Show that the following are equivalent: i. $R$ is a division ring. ii. $R$ is nontrivial and if $r$,$s \in R$, with $rs=0$, then either $r=0$ or $s=0$. $\textbf{NOTE:}$ A ...
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### On the opposite group

When the group $G$ acts on the set $S$, e.g. $G \colon= {\mathrm{GL}}_2({\Bbb C})$, $S \colon= {\Bbb C}^{\oplus 2}$, $G$ have the right to act on the left or on the right. Both actions can be ...
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### A finite ring without zero divisors is a division ring [duplicate]

Question: Let R be a finite ring without zero divisors and |R| > 1. Then show that R is a division ring. I used the following logic, but I'm not sure if this is correct. Let |R| = n, a finite ...
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### Analogue of primitive element theorem for division rings

I was wondering, is the following analogue of the primitive element theorem true for division rings: Let $R,S$ be division rings of characteristic $0$ such that $R$ is finite dimensional as an $S$-...
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### Let $F$ be an infinite field and let $f(x) ∈ F[x]$. If $f(a) = 0$ for infinitely many $a ∈ F$, show that $f = 0$. [duplicate]

Step 1: Suppose that $F$ is an infinite field and $f(x) \in F[x]$. To claim the statement, "If $f(a)=0$ for infinitely many elements $a$ of $F$, then $f(x)=0$". To prove this statement using ...
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### Show that $\mathbb{Z}[\sqrt{3}]$ satisfies the following division property:

Here is the question I want to solve: Show that $\mathbb{Z}[\sqrt{3}]$ satisfies the following division property: Given $a,b \in \mathbb{Z}[\sqrt{3}]$ there exist $q,r \in \mathbb{Z}[\sqrt{3}]$ such ...
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### Must an abstract $C^*$-algebra that is an integral domain be a field?

I'd like to understand a little bit better why the maximal (as opposed to prime) spectrum is the appropriate notion of spectrum for the theory of $C^*$-algebras. The canonical answer is that $C^*$-...
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### Division ring and Quadratic Extension

First We define involution map which is map $*$ : $R \rightarrow R$ such that it satisfy $i)\ (a+b)^* = a^*+ b^*$ $ii)\ (ab)^* = b^*a^*$ $iii) (a^*)^* = a$ for $a,b$ in $R$. Now $Z$ be center ...
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### Definition of a division ring in category theory

I'm wondering how one can define a division ring in category theory. More precisely, is there a well-defined concept of "division ring object" such that a division ring object in the category of sets ...
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### Algebraic structure, division ring

In the set $\mathbb{R}\times \mathbb{R}^3$ addition is defined by components. We define multiplication $*$ by (\lambda,\mathbf{x})*(\mu,\mathbf{y})=(\lambda\mu - \mathbf{x}\cdot \mathbf{y}, \lambda \...
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### How to complete the whole without fractions [closed]

Let's say I need to split the following whole number 4 ways: 1659. I cannot have any remainders, yet I need the sum of the four divisors to equal the sum of the dividend. Is there any mathematical ...
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### Every left ideal of $R[x]$ is cyclic implies $R$ a division ring

I need to prove that if every left ideal of $R[x]$ is cyclic as a left $R[x]$-module then $R$ is a division ring. I don't really understand. So a left ideal $I$ is cyclic as a left $R[x]$-module, let ...
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### Injective module

Let $R=M_n(D)$ the Matrix ring over a division ring and consider $R$ as a left module over itself. Is $R$ an injective module? I know that $R$ is free, hence it is projective. Is $R$ injective? I ...
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### Splitting fields of a divison algebra

Let $k$ be a field, $D$ be a central division algebra of degree $n$ over $k$. We call $k'$ a splitting field of $D$ if $D\otimes_kk'\cong M_n(k')$. Splitting fields may not be isomorphic, can we say ...
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### 2-digit number Modulus Nine.

There are two distinct 2-digit numbers which have the same units digit but different tens digits. The quotient when one of them is divided by 9 is equal to the remainder when the other is divided by 9,...
By a left simple ring I mean a ring with no proper, nontrivial left ideal. $R$ be such a ring. Let $u(≠0)\in R$. Then $Ru=R$ (since $1u=u≠0$). Now $Ru=${$ru:r\in R$}. So there is an $r\in R$ such ...
I have a (D,F)-bimodule $A$ where $D,F$ are division rings and $A = aF$ for some $a\in A$. then for every $d\in D$, $da\in A$ so there exists a unique $f_d\in F$ such that $da = af_d$. Then I define ...