ideal classes can be multiplied: if ["i"] denotes the equivalence class of the ideal "i", then the multiplication ["i"]["j"] = ["ij"] is well-defined and commutative. the principal ideals form the ideal class ["r"] which serves as an identity element for this multiplication. thus a class ["i"] has an inverse ["j"] if and only if there is an ideal "j" such that "ij" is a principal ideal. in general, such a "j" may not exist and consequently the set of ideal classes of "r" may only be a monoid.