# 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: (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 ...
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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 ...
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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 ...
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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 ...
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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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### 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 ...
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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. ...
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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?...
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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 ...
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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 $c\in ... • 705 0 votes 0 answers 99 views ### How can I create the presentation (min. set of relations) of the quaternion group of order 8? How can I look for conjugation relations? What are the steps that leads me to know the presentation (specifically defining the generators and the minimum set of relations required to define this group)of the quaternion group of order 8? What ... • 989 0 votes 0 answers 372 views ### when does a quaternion algebra isomorphic to$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$B$... • 705 3 votes 1 answer 101 views ### 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 ... 1 vote 1 answer 120 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 ... • 2,471 0 votes 0 answers 39 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 ... • 705 0 votes 1 answer 78 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 ... • 705 1 vote 2 answers 116 views ### 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 ... • 2,808 1 vote 1 answer 67 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,... • 705 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 ... • 1,554 1 vote 1 answer 151 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 ... 1 vote 0 answers 130 views ### 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 ... -1 votes 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 ... -2 votes 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 ... 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 ... 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 ... • 191 0 votes 0 answers 84 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$S$-... 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 ... 0 votes 0 answers 69 views ### 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 ... 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$C^*$-... • 9,897 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 \...
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 ...