Commutator of a ring

Summary

The commutator of two elements $x, y$ of a ring or algebra is $[x, y] = xy - yx$. It is zero exactly when $x$ and $y$ commute, so it measures the failure of commutativity. In an associative algebra the commutator is a Lie bracket, and this turns every associative algebra into a Lie algebra. It is the ring analogue of the commutator of two group elements.