Theorems about abelian groups (i.e. modules over the principal ideal domain formula_54) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form formula_72 for formula_73 prime, and the latter is a direct sum of finitely many copies of formula_54.