a "ring" likewise, has a set of elements, but now has "two" operations. the first operation must make the set a group, and the second operation is associative and distributive with respect to the first operation. it may or may not be commutative; this means that the result of applying the operation to a first and a second element is the same as to the second and first – the order of the elements does not matter. if every non-zero element has a multiplicative inverse (an element "x" such that "a x" = "x a" = 1 ), the ring is called a "division ring". a "field" is defined as a commutative division ring.