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.