Questions tagged [division-ring]

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

Filter by
Sorted by
Tagged with
0
votes
1answer
19 views

$ab=0 \Rightarrow b$ 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 ...
0
votes
0answers
28 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 ...
2
votes
1answer
30 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 ...
-2
votes
1answer
18 views

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?
0
votes
1answer
21 views

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?
2
votes
1answer
18 views

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 $...
0
votes
0answers
17 views

Analysis of the Adjoint representation of Lie Algebra

I'm attempting to find $f\left(ad_{x}\right)y$ where $ad_{x}y=xy-yx$ for any function f and rings x,y. adjoint representation I have a way to calculate it for any holomorphic function. However, I'm ...
1
vote
1answer
64 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{...
5
votes
0answers
67 views

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 ...
1
vote
0answers
39 views

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 ...
6
votes
0answers
114 views

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 ...
2
votes
1answer
62 views

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 $4$-...
1
vote
0answers
34 views

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 ...
1
vote
1answer
41 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.
1
vote
1answer
28 views

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 ...
0
votes
0answers
16 views

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? ...
2
votes
0answers
27 views

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 ...
5
votes
0answers
81 views

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 ...
1
vote
0answers
13 views

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 $\...
0
votes
0answers
39 views

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 ...
5
votes
1answer
90 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 ...
2
votes
2answers
65 views

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 \...
0
votes
1answer
44 views

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 ...
1
vote
0answers
56 views

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 ...
2
votes
2answers
68 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 ...
2
votes
1answer
110 views

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 ...
0
votes
1answer
73 views

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,...
0
votes
0answers
13 views

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 ...
1
vote
0answers
32 views

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 ...
1
vote
1answer
17 views

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 $E$ ...
10
votes
1answer
117 views

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 ?
-1
votes
1answer
40 views

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 ...
2
votes
0answers
32 views

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 $\pi$-...
5
votes
1answer
166 views

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 ...
3
votes
0answers
83 views

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 ...
1
vote
1answer
235 views

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'...
2
votes
0answers
92 views

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 $D$-...
-1
votes
1answer
167 views

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 ($D=Z(D)$).
1
vote
1answer
1k 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. $...
9
votes
1answer
100 views

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, ...
4
votes
1answer
271 views

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 ...
3
votes
0answers
150 views

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 $R_2\subset R$ ?
3
votes
0answers
169 views

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 (...
3
votes
1answer
49 views

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 \...
2
votes
0answers
58 views

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 $\...
3
votes
1answer
35 views

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 ...
0
votes
2answers
92 views

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\, \...
3
votes
2answers
129 views

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 ...
1
vote
2answers
2k views

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.