How to proof there is no idempotent element other than 0 and 1 in a Division Algebra?

Question

If $A$ is a division $K$-algebra. Then I need to proof there is no idempotent element other than $0$ and $1_A$ in $A$. I tried this way : If $0,1_A\neq a\in A$ such that $a^2=a.~$ Now $A$ is division algebra and $a\neq 0$, so $\exists b\in A$ such that $ab=ba=1_A$. So $a^2=a\Rightarrow a^2b=ab\Rightarrow a(ab)=1_A\Rightarrow a1_A=1_A.~~~\therefore a=1_A$. Hence contradiction. But the problem is for doing this $A$ has to be associative. How can I prove it in general?

Answer 1

I'm not up on division algebras, but the Wikipedia definition is

We call $D$ a division algebra if for any element $u$ in D and any non-zero element $v$ in $D$ there exists precisely one element $x$ in $D$ with $u = vx$ ...

If $a$ is an idempotent, then

$$ 0 = a^2 - a = a(a- 1_A)$$

And of course we know that $ 0 = a \cdot 0$.

So if we assume $a \ne 0$, the division algebra uniqueness gives us $ a - 1_A = 0$.