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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28 views

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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34 views

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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66 views

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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51 views

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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90 views

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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51 views

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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53 views

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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163 views

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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56 views

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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257 views

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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63 views

Find a way to represent $\mathbb{H}$ as a subring of $M_{4}(\mathbb{R}).$

Here is the question that I want to answer part(c) in it: Define $E \in GL_{2}(\mathbb{R})$ by $E = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and let $\mathcal{R} = \{aI + bE| a,b \in \...
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Division in higher dimensional ring of formal power series

Let $R$ be a ring with characteristic $0$. We can assume $R$ to be the $p$-adic field also, which is of course characteristic $0$. Let $f(x) \in R[[x]]$ be a power series and $f^{k}(x)$ be its $k$-th ...
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$ab=0 \Rightarrow b$ is nilpotent in a Noetherian ring

Let $A$ be a Noetherian ring and $a \in A$. Show that if $a$ is not contained in a minimal prime, $ab=0 \Rightarrow b$ is nilpotent. I can't see a way to solve this. I tried to consider that in a ...
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32 views

Contain does not imply divide

Let $\mathbb{Z}[X]$ be the ring of polynomials over a single variable $X$. Let $(2)\subseteq(2,X)$ be ideals of $\mathbb{Z}[X]$. I want to prove that it is impossible to write the ideal $(2)$ as a ...
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37 views

Uncommon notation for division algebra

I have found the following notion for a division algebra in a paper. $K=\mathbb{R}(x_1, \dots, x_n)$ is the field of rational functions in $n$ variables over $\mathbb{R}$ and $F=K((t))$ be the field ...
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Always exist a linear transformation from an infinite dimensional vector space X to X that is a surjective but injective?

In the ring of all linear transformation of an infinite dimensional vector space over a division ring, always exist a linear transformation which is surjective but no injective?
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A question about inverse square matrix on division ring. [closed]

Does every square matrix on a division ring which has a left inverse have also a right inverse?
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Can we make any conclusion on the numerators of elements in the fraction division ring $D(x)$?

Let $D$ be a division ring. Let $D[x]$ be a polynomial ring in a central indeterminate $x$. And let $D(x)$ be the quotient ring of fraction of $D[x]$. So, let $f,g \in D[x]$ such that $g \neq 0$ and $...
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117 views

Show that $\mathbb{H}$ is a division ring

Let $\mathbb{H} \subset M_2(\mathbb{C})$ be the set of matrices of the form: $$A = \begin{pmatrix}z & -\bar{w}\\w & \bar{z}\end{pmatrix} \text{ where } z,w \in \mathbb{C}$$ $A^{-1} = \begin{...
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Proving Division Rings of $p^2$ Elements are Fields

Exercise III.2.11 (Aluffi, Algebra Ch 0): Let $R$ be a division ring consisting of $p^2$ elements, where $p$ is a prime. Prove that $R$ is commutative (and thus $R$ is a field). Note: I do so without ...
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if $S$ is a ring with no proper left ideals, then either $S^2=0$ or $S$ is a division ring. [duplicate]

I am reading Hungerford's Algebra in Chapter 3(Rings), and I am stuck on the following question. if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is ...
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Reflection groups of division rings

My question is: Is there a classification of finite groups representable as a $\mathbb{K}$-reflection group for some division ring $\mathbb{K}$ of characteristic zero? I would also appreciate any ...
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Quaternion algebra: $a \equiv 3$ or $5\mod 8$ implies $(a,2)_{\mathbb{Q}}$ is a division ring.

Question: let $a\in\mathbb{Z}$ such that $a\equiv 3$ or $5\mod 8$. Proof that $(a,2)_{\mathbb{Q}}$ is a division ring. Definition: a quaternion algebra over a field $F$ is a ring that is a $4$-...
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Can the division rings appearing in the Wedderburn-Artin theorem be isomorphic?

I am asking because I was thinking about this problem: Can we determine the simple modules of a semisimple ring from its product decomposition given by the Wedderburn-Artin theorem? Let me state ...
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63 views

If $\mathbb{k}$ is a division ring then $\mathbb{k}^n$ is a simple $M_n(\mathbb{k})$ module

Problem: If $\mathbb{k}$ is a division ring then $\mathbb{k}^n$ is a simple $M_n(\mathbb{k})$ module I'm lost on this problem, the hint is to use linear algebra but i dont see how it helps.
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Modulus transformation

Can someone confirm whether this is true : $((a^b \ mod\ n) * (a^c \ mod \ n)) \mod \ n = a^{b+c} \ mod \ n$ Im pretty sure it is, and every combination of numbers i try manually works, but when i ...
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Ringoid over symmetric group? (part 2)

Last time, it turned out that there is a near-ring whose domain and additive operation is that of symmetric groups. What if the operation is taken as multiplicative instead? Could it be a skew field? ...
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Does every Archimedian partially ordered division ring embed in the reals?

I know that any Archimedian totally ordered field embeds into the real numbers. Does this result extend to a priori partially ordered field? A division ring is basically a noncommutative field. Does ...
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What is the intuition behind Hua’s proof of the Cartan-Brauer-Hua theorem?

The Cartan-Brauer-Hua theorem states that Let $K\subset D$ be division rings so that whenever $x\in D$ is a nonzero element, $xKx^{-1}\subset K$. Show that either $K\subset Z(D)$ or $K=D$. This ...
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Finite generation of ring of automorphisms

$\DeclareMathOperator{\Frac}{Frac}$ Hello everybody. My question is about Proposition $6.2.7.$ in the book "Basic Structures of Function Field Arithmetic" by David Goss. Given a field $L$ with an $\...
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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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132 views

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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109 views

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,...
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Is a left and right simple ring with unity a division ring? [duplicate]

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 ...
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isomorphism between two division rings

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 ...