Find a concept
Start typing to search the mathematical index.
Over a PID, finitely generated torsion-free modules are free.
Finitely generated torsion-free over a PID is free: Let be a PID and let be a finitely generated torsion-free -module. Then is a free -module.
This is the torsion-free specialization of the structure theorem for finitely generated modules over a PID: for a PID, a finitely generated torsion-free module has no torsion summand and must therefore be free.
Structure theorem (f.g. modules over a PID): Let be a principal ideal domain and let be a finitely generated module over . Then there exist an integer and nonzero elements with such that
a decomposition as a direct sum of a free part and cyclic torsion factors. The integer and the invariant factors are unique up to multiplication by units in . In particular, the torsion submodule of is a torsion module and the free summand is a free module of rank .
This theorem is the module-theoretic engine behind the classification of finitely generated abelian groups (the case ); see classification of finitely generated abelian groups. It is closely tied to Smith normal form, and it admits an equivalent primary decomposition form (see the elementary divisor theorem).
A principal ideal domain (PID) is an integral domain such that every ideal is principal, i.e. for some .
PIDs provide strong control of divisibility and modules, and they sit between Euclidean domains and unique factorization: every Euclidean domain is a PID, and every PID is a UFD (see PID implies UFD).
A left -module is finitely generated if there exist elements such that every can be written as
for some .
Finiteness hypotheses are central for structure theorems (e.g. over a PID) and for Noetherian conditions.
Let be an integral domain and an -module. The module is torsion-free if its only torsion element is , i.e. if with forces .
Torsion-freeness is weaker than freeness but is a key hypothesis in many classification results over PIDs and in the theory of lattices.
A left -module is free if it has a basis , i.e. every element of admits a unique expression as a finite -linear combination of elements of .
Free modules are characterized by the universal property of free modules: functions from a basis extend uniquely to module homomorphisms. Over a field, free modules coincide with vector spaces and their bases agree with linear-algebra bases.