1. ## commutator ring

let R be ring, [a,b] is called commutator if
[a,b]=ab-ba for every a,b in R

any example for this?

i read some condition in derivation algebra (every jordan is derivation) says "have commutator which is not right zero divisor"
is that possible? cause commutator always have zero element (in my opinion)
[a,a]=0

2. ## Re: commutator ring

Hmm? I'm not 100% sure what you are trying to say/ask. The commutator is an operation defined on pairs of elements a ring, just like addition and multiplication are.

Let $\displaystyle R$ be a ring, and let $\displaystyle a,b\in R$. The commutator of $\displaystyle a$ and $\displaystyle b$ is defined to be $\displaystyle [a,b]=ab-ba$.

You can define this in absolutely any ring. Of course, it may be more useful in some rings than in others, depending on what you are trying to do.

3. ## Re: commutator ring

then commutator is set right?
if R is commutative ring, the commutator is set of zero elements?

i'm trying ask statement "Ring has a commutator which is not a zero divisor"
zero divisor here is all element of commutator or some elements?
(cause commutator always have zero elements which is zero divisor)