The group of integers Z under addition, denoted ⁠ ( Z , + ) ⁠, has been described above. The integers, with the operation of multiplication instead of addition, ( Z , ⋅ ⋅ ) do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a ⋅ ⋅ b = 1 in this case is ⁠ b = 1 2 ⁠, which is a rational number, but not an integer. Hence not every element of Z has a (multiplicative) inverse.