commutator -
Of a group an element of the form ghg−1h−1 where g and h are elements of the group; it is equal to the group's identity if and only if g and h commute.
commutator -
Of a ring an element of the form ab-ba, where a and b are elements of the ring, it is identical to the ring's zeroelement if and only if a and b commute.
Wikipedia
The commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
