Questions tagged [division-ring]

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

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Triple Cross Product Identity for imaginary quaternions $\mathbb{H}_0$

Considering $x,y,z \in \mathbb{H}_0$, $x,y,z=\alpha$i $+ \beta $j $+ \gamma $k, prove the Triple Cross Product Identity: \begin{equation} (x \times y) \times z = y(x \bullet z) - x(y \bullet z) \end{...
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A finite division ring $D$ is a field

7.8.12 Wedderburn: A finite division ring $D$ is a field. I have understood several theorems from this book (A first course in abstract algebra by Hiram, paley) by my own, but this one is beyond ...
N00BMaster's user avatar
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A special case of Skolem-Nother 'sTheorem

Assume that $D$ is a division ring and $n>1$ be a natural number. Let $a\in SL_n(D)$ be a torsion element. For example, $a^m=1$. Also consider that $F=Z(D)$. Therefore, $[F[a]:F]<\infty$. By ...
Reza Fallah Moghaddam's user avatar
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Centre of fixed skew field is the fixed field of centre?

Let $D$ be a skew field, assumed to be finite dimensional over its centre $Z(D)$. Let $\sigma\in\mathrm{Aut} (D)$ be an automorphism of $D$, and let $D^{\sigma}$ be the set of elements in $D$ fixed by ...
Hermetically Sealed Halibut's user avatar
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Show that a division ring contains exactly two idempotent elements.

My proof: Suppose that $R$ is a division ring. Since $0a=a0=0$, then $0^2 =0$. If $a \neq 0$ and $a^2 = a$, then the inverse $a^{-1}$ of $a$ exists. So, $a^{-1} (a^2)=a^{-1} (a)$. This implies that $a=...
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Let $D$ a division ring. Why $D^n$ needs to be a right $D$-vector space?

Let $D$ be a division ring then take $M_n(D)$ the matrix ring over $D$. Now we have that $D^n$ the only simple left $M_n(D)$-module. During the proof some authors (e.g. Lam in "A first course in ...
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On the property of a ring modulo its Jacobson radical being a division ring

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $A$ be a finitely generated associative $R$-algebra. Let $x\in \mathfrak m$ be a non-zero-divisor on $A$ such that $xA\neq A$. If $A/J(...
uno's user avatar
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Prove that the ring of real quaternions modulo $p$ is not a division ring.

Using the ring of real quaternions as a model, we define the quaternions over the integers $\text{mod}$ $p,$ $p$ an odd prime number, in exactly the same way; however, now considering all symbols of ...
Thomas Finley's user avatar
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Wedderburn's little theorem

I'm trying to understand the proof of Wedderburn's theorem which states that every finite division ring is a field. I'm following the proof given by Herstein in his book Algebra. Wedderburn's theorem ...
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Division ring of Quaternions [closed]

Let $H$ be the Division ring of Quaternions. Let $\alpha =a + bi +cj + dk \in H$, we will call $\bar{\alpha} = a - bi - cj - dk$ the conjugate of $\alpha$. Denote $Tr(\alpha) := \alpha + \bar{\alpha} =...
Na Man's user avatar
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Problem regarding an associative ring being a division ring. [duplicate]

I was trying to solve a problem in C. Musili's book "Introduction to Rings and Modules" and I got stuck while trying to solve the following problem: Let $R$ be a non-zero ring such that the ...
Sharang Thimmaiah's user avatar
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Left invertible matrix over a division ring [duplicate]

Is it true that every left invertible square matrix Α over a division ring R is also right invertible? If that is the case and B is the left inverse of A and C is the right inverse of A , then B=C? ...
Paranoi_D's user avatar
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Why is $D$ of characteristic $p \gt 0$ in this proof?

From Section $7.2$ of Herstein's "Topics in Algebra" ($2^{\text{nd}}$ edition): THEOREM $7.2.2$ (JACOBSON) $\;$ Let $D$ be a division ring such that for every $a \in D$ there exists a ...
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The multiplicative group of the real quaternion division ring

Let $\mathbb{H}$ be the real quaternion division ring, that is, $\mathbb{H}$ consists of all elements of the form: $a+bi+cj+dk$ in which $a,b,c,d\in\mathbb{R}$ and $i^2=j^2=k^2=-1,ij=-ji=k$ with usual ...
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For quaternions $\alpha$ and $\beta$, find all quaternions $x$ such that $\alpha x=x\beta$.

Let $\mathbb{H}$ be the real quaternion division ring consisting of all elements of form: $a+bi+cj+dk$ in which $a,b,c,d\in\mathbb{R},i^2=j^2=k^2=-1, ij=-ji=k$ and $\alpha=a+bi+cj+dk,\beta=a+\sqrt{b^...
Tran Nam Son's user avatar
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Classification of graded fields, semifields, and skew fields

Recently I came upon the following result: Result 1. Let $K$ be a $\mathbb{Z}$-graded field. Then either $K$ is trivially graded (i.e. $K_k=0$ for $k\in\mathbb{Z}\setminus\{0\}$ with $K_0$ a field or ...
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Is every left primitive ring a division ring?

I was solving some exercises about left primitive rings but the "proof" I found for them doesn't use all the assumptions, so I wanted to know if there are left primitive rings wich are not ...
Diogo Santos's user avatar
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Show that if $R$ is a division ring and $S$ is a subring of $R$ then $S$ is also a division ring.

I'm trying to do problem 7.1.7 in Dummit and Foote. Problem: Prove that the centre of a ring $R$ is a subring of $R$ that contains the identity. Prove that the center of a division ring is a field. ...
Irving Rabin's user avatar
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Matrix Ring over a Division Ring

What is the definition of a matrix ring over a division ring? Hungerford mentions this in one of his examples but does not define what it is. Is it simply a matrix ring with entries of a division ring?...
Kenneth Winters's user avatar
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Classification of "complete dense ordered near-semirings"

Let's define a complete dense ordered near-semiring, or a CDON, as a set $A$ equipped with two binary operations $+,\times$ and a binary relation $\leq$ such that: $+$ is a monoid, whose identity is ...
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When is $R/M$ a division ring?

There is a famous theorem in commutative ring theory which states: "Let $R$ be a commutative ring with unity and let $M$ be a (two-sided) ideal of $R$. Then, $M$ is maximal if and only if $R/M$ ...
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Why is $M_n(D)$ a right quotient ring?

Let $n$ be an interger and $D$ a division ring. I want to understand why $R=M_n(D)$ (the ring of $n\times n$ matrices with entries in $D$) is a right quotient ring (or a classical quotient ring). I ...
William's user avatar
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For distinct primes p and q that are $3$ mod $4$, $(−1, p)_\mathbb{Q}$ is not isomorphic to $(−1, q)_\mathbb{Q}$.

For distinct primes p and q that are $3$ mod $4$, the quaternion algebras over the rationals, $(−1, p)_\mathbb{Q}$ is not isomorphic to $(−1, q)_\mathbb{Q}$. Any help prove it? I'm able to show that $...
MZG's user avatar
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A semisimple ring with a finite number of left maximal ideals

Let $R$ be a semisimple ring with a finite number of left maximal ideals. (Here "semisimple" means that the Jacobson radical is zero.) Show that $R \cong R_1 \times ... \times R_n$ Such ...
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Do matrices over noncommutative division rings have well-defined ranks?

It is known that the row and column rank of any matrix over a field are the same and their common value is simply called the rank of the matrix. Now, for any $m$-by-$n$ matrix $A$ with entries in a ...
Geoffrey Trang's user avatar
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1 answer
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Showing that End$_{FG}V$ is isomorphic to $F$ as a ring.

Let $G$ be a finite group, $F$ be an algebraically closed field, and $V$ an irreducible $FG$-module. I want to show that End$_{FG}V$ is isomorphic to $F$ as a ring. I managed to get that End$_{FG}V$ ...
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1 answer
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Understanding a proof of Wedderburn in Topics in algebra

I am working on the proof of Wedderburn theorem and I have a problem to understand the part of it. In Herstein's book, I can't understand the claim "$r$ is a prime number." I am not clear ...
WHERE 234's user avatar
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Skew fields with nonzero characteristic

There are many ways to construct a skew field of nonzero characteristic, e.g. the universal field of fractions of a skew polynomial ring $E[x;\sigma]$, a suitable choice of a quaternion algebra over $...
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The quadratic equation $x^2=c$ in a division ring

Let $D$ be a division ring. We denote $D'$ by the derived subgroup of the multiplicative group $D\setminus\{0\}$, that is, the subgroup generated by all the commutators of $D\setminus\{0\}$. For $c\in ...
Tran Nam Son's user avatar
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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 ...
Brain's user avatar
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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 $B$ ...
Tran Nam Son's user avatar
3 votes
1 answer
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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 ...
Dr. Nirbhay Kumar's user avatar
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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 ...
Tanny Sieben's user avatar
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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 ...
Tran Nam Son's user avatar
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1 answer
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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 ...
Tran Nam Son's user avatar
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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,...
Tran Nam Son's user avatar
3 votes
1 answer
153 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 ...
tox123's user avatar
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1 vote
1 answer
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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 ...
Promit Mukherjee's user avatar
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2 answers
992 views

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 ...
hydthemoon's user avatar
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3 answers
1k views

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 ...
Doodle Bob's user avatar
1 vote
1 answer
118 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 ...
Pierre MATSUMI's user avatar
2 votes
2 answers
1k 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 ...
user1742188's user avatar
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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 $S$-...
Suchir Kaustav's user avatar
2 votes
1 answer
989 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 ...
Sir Munda's user avatar
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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 ...
user avatar
3 votes
1 answer
172 views

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 $C^*$-...
Jacob Manaker's user avatar
2 votes
1 answer
144 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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2 votes
0 answers
59 views

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 ...
MAS's user avatar
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