Structure theorem for finitely generated modules over a principal ideal domain

Every finitely generated module over a principal ideal domain is a direct sum of a free module and finitely many cyclic torsion modules. The free rank and torsion invariant factors are uniquely determined up to isomorphism.

Smith normal form: A diagonal form for integer or principal-ideal-domain matrices, with each diagonal entry dividing the next. It diagonalizes a presentation matrix and exposes the module’s invariant factors.
Principal ideal domain: An integral domain in which every ideal is generated by one element. Its principal ideals enable the divisibility reductions used to classify modules.
Finitely generated abelian group: An abelian group generated by finitely many elements. Taking the ring to be the integers recovers the classification of these groups.
Vector space: A module over a field, with every nonzero scalar invertible and every module free. Over a field, torsion summands disappear, leaving only dimension.
Module presentation: A description of a module as a quotient of a free module by a submodule of relations. The theorem is obtained by simplifying the relations in a finite presentation.
Finitely generated module: A module whose elements are generated by a finite set under addition and scalar multiplication. Finite generation ensures a finite presentation and only finitely many cyclic summands.
Classification of finite abelian groups: The theorem that every finite abelian group is a direct sum of cyclic groups of prime-power order. This is the theorem’s integer-ring specialization for modules with zero free rank.
Module over a non-principal ring: A module over a ring that has at least one ideal not generated by a single element. Without principal ideals, finitely generated modules need not have this direct-sum classification.
Invariant factor decomposition: A decomposition of a finitely generated torsion module into cyclic modules whose orders form a divisibility chain. Its divisibility-ordered cyclic summands give one canonical form in the theorem.
Cyclic module: A module generated by one element, isomorphic to a quotient of the ring by an ideal. Each torsion summand in the theorem is cyclic.

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Wedderburn–Artin theoremCompared with: It is another algebraic classification theorem, but concerns modules over a different class of rings.