# Questions tagged [division-ring]

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

31 questions
0answers
30 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 ...
1answer
75 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 ...
2answers
44 views

2answers
118 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 ...
2answers
1k 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.