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Questions tagged [division-ring]

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

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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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Seminorm on a division ring

Suppose that $D$ is a division ring carrying a non-archimedian seminorm $\vert\cdot\vert:D\to\mathbb{R}$, i.e. $\vert a+b\vert\leq\max\{\vert a\vert,\vert b\vert\}$ and $\vert ab\vert\leq \vert a\vert\...
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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,...
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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 ...
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division ring of left multiplications

I have a division ring $D$, $A$ a ring of endomorphisms of the additive group of $D$ and define the set and $G$ a subring of $A$ containing the right multiplications by elements of $D$. I define $E$ ...
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If $D$ be a division ring and $D^*$ be finitely generated group then $D^*$ is abelian group?

Wedderburn's little theorem : every finite division ring $D$ is commutative, or $D^*$ is abelian group. Now if $D^*$ be a finitely generated group then $D^*$ is an abelian group ?
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Is factor ring field? [duplicate]

There is defined Z[i] = {a+bi|a,b $\in$ Z} with standard operations of addition and multiplications complex number. Question is, if factor ring Z[i]/(1-i) is field. How could I prove it? Do you ...
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Growth rates of matrices

Let $D$ be a discrete valuation ring, possibly non-commutative, with uniformiser $\pi$, and let $Q=M_n(Q(D))$, where $Q(D)$ denotes the ring of quotients of $D$. Let $v$ be the extension of the $\pi$-...
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Galois theory for (non-commutative) division rings

Is there a 'Galois theory' with fields replaced by (non-commutative) division rings? I have googled this, and it seems that there are known results in that direction, for example, this paper which ...
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Difficulties with “old” definitions

I have a paper by Jacobson Nathan Jacobson.Structure of Rings, Volume 37, Part 1. American Math-ematical Soc., revised edition, 1956. Which really uses definitions that seem very complicated for ...
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1answer
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Prove that for each submodule $B$, there exist a submodule $C$ such that $A=B\oplus C$.

Problem: Let $A$ be a unitary module over a division ring $R$. Prove that for each submodule $B$, there exist a submodule $C$ such that $A=B\oplus C$. Anyone can help me in this problem? I really don'...
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If $R$ is a finitely generated $D$-algebra, $D$ a division ring, then $R$ has finite basis over $D$?

Let $k \subset D \subset R$, where $k$ is a field of characteristic zero, $D$ is a division $k$-algebra, and $R$ is affine over $D$ (= $R$ is a finitely generated $D$-algebra). $R$ is a free $D$-...
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Let $D$ be a division ring. Show that if every $a \in D$ is algebraic over the prime subfield of $D$ then $D$ is commutative [closed]

Let $D$ be a division ring. Show that if every $a \in D$ is algebraic over the prime subfield of $D$ then $D$ is commutative ($D=Z(D)$).
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1answer
627 views

Every simple ring is a division ring and vice versa.

My definition of simple ring is: A ring $R$ where $\{0\}$ is a maximal ideal, is called a simple ring. Now, assume $R$ is not trivial. Wherever I write $I \subset R$, I mean that $I$ is an ideal. $...
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Abelianisation of a division ring

I was reading a paper recently concerning a non-commutative version of the matrix determinant. On the third page, it stated a fact without providing a proof or a reference: If $D$ is a division ring, ...
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1answer
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Proof of brauer's lemma, $eRe$ being a division ring.

On page 1 of this article, the author proves the following claim: Brauer's Lemma: Let $K$ be a minimal left ideal of a ring $R$, with $K^2 \not= 0$. Then $K=Re$ where $e^2=e \in R$, and $eRe$ is a ...
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Can one embed two division rings in a common one?

Given two division rings $R_1,R_2$ having the same characteristic, is there a simple way to find a division ring $R$ with two embedings $R_1\subset R$ and $R_2\subset R$ ?
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The division ring of fractions of the first Weyl algebra and its subrings

The first Weyl algebra, $A_1(k)= k\langle x,y | yx-xy=1\rangle$, where $k$ is a field of characteristic zero, is known to be a simple Noetherian ring, hence it has a (left) division ring of fractions (...
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1answer
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Multiplicativity formula for a tower of division algebras

The multiplicativity formula for degrees of a tower of fields is well-known. I wonder if the same formula still holds if we consider division rings instead of fields, namely: Let $A \subseteq B \...
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Let $R = \mathbb{Q}[x]$ and let $I = (x^2 + 2x + 2)R$ be the principal ideal generated by $x^2 + 2x +2$. Two questions are below.

i) Show that any element of $R$ is congruent modulo $I$ to a unique polynomial of the form $ax+b$ where $a,b \in \mathbb{Q}$? ii) Show that any element of the quotient ring $R/I$ is of the form $\...
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A proper subring of an 4-dimensional division F-algebra is a field.

I'm stuck with this thing: Let $F$ be a field of characteristic $\neq 2$, let $D$ be a 4 dimensional noncommutative division algebra over $F$. For $x\in D\smallsetminus F, F[x]$ is a field of ...
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Homomorphic images of nth power of a division ring

Let $R$ be a division ring and $n \geq 2$. I need to identify up to isomorphism all homomorphic images of the $n$-th power of $R$: $$ R^{n} = R \oplus R \oplus \cdots \oplus R\quad (\text{for}\,n\, \...
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Is $R[i,j,k]$ a division ring whenever $R$ is a field?

Let $R$ be a field , let us adjoin $i,j,k$ to $R$ and write $$R[i,j,k]:=\{a+bi+cj+dk :a,b,c,d \in R \}$$ where $i,j,k$ satisfies $a+bi+cj+dk=0 $ iff $a=b=c=d=0$, $i,j,k$ commutes with every element of ...
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Is every simple ring a division ring?

I know that every division ring is simple. Is the converse true? I think it isn't. But I can't find a counterexample.