Additive commutator
Definition
Suppose is a non-associative ring (i.e., a not necessarily associative ring) with binary operation . The additive commutator, sometimes simply called the commutator on is a function:
![{\displaystyle [\ ,\ ]:R\times R\to R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6377efb3611ccbd76e3487da51846ca1b45e183)
defined as:
![{\displaystyle [x,y]:=(x*y)-(y*x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd66def1bbef77c19d5d2dc2a4e51b44f1209bf8)
Note that the term commutator in group theory is used for the group commutator, which is a different but related notion. If we are dealing with a ring and with subgroups of the multiplicative group, the terminology may be confusing. In this case, we use multiplicative commutator for the group commutator and additive commutator for the commutator as defined above.
Facts
- If is an associative ring, then the additive commutator on its defines a Lie ring structure. This is termed the associated Lie ring of an associative ring.
- If is an alternative ring, then the additive commutator on its defines a Malcev ring structure. This is termed the associated Malcev ring of an alternative ring.