the integers form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. any pair of integers can be added or multiplied, always resulting in another integer, and the first operation, addition, is commutative, i.e., for any elements "a" and "b" in the ring, "a" + "b" = "b" + "a". the second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that "a" combined with "b" might be different from "b" combined with "a". examples of noncommutative rings include matrices and quaternions. the integers do not form a division ring, because the second operation cannot always be inverted; there is no integer "a" such that 3 × "a" = 1.