# Questions tagged [division-ring]

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

49 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 ...
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?
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?
18 views

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 ...
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 ...
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 ...
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 $4$-...
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 ...
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.
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 ...
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? ...
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 ...
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 ...