Free Module

A free module is a module that admits a basis, the closest analogue of a vector space in module theory and the universal building block from which all modules are quotients.

Definition

A free module over a ring is a module isomorphic to a direct sum of copies of the ring, equivalently a module possessing a basis, a linearly independent generating set.

Scope

This topic covers the definition of a free module, its universal property, rank and the invariant dimension property for commutative rings, the presentation of arbitrary modules as quotients of free modules, and the related notion of projective modules.

Core questions

• What does it mean for a module to have a basis?
• What universal property characterizes free modules?
• Is the rank of a free module well defined?
• How does every module arise as a quotient of a free module?

Key theories

Universal property of free modules

A free module on a set is universal among modules receiving a map from that set: any function from the basis to a module extends uniquely to a module homomorphism, making free modules the free objects of module theory.

Invariance of rank

Over a commutative ring with identity, any two bases of a free module have the same cardinality, so the rank is a well-defined invariant, generalizing the invariance of dimension for vector spaces.

Free presentations

Every module is a quotient of a free module by a submodule of relations, giving a presentation by generators and relations; when the relation module is also free this is a free resolution, the start of homological algebra.

Clinical relevance

Free modules are the workhorse of computational and homological algebra: presentations and resolutions by free modules compute invariants such as Tor and Ext, and over principal-ideal domains the interplay between free and torsion submodules yields the structure theorem behind canonical forms and the classification of abelian groups.

History

The notion of a basis for a module generalized the bases of vector spaces and the free abelian groups of nineteenth-century arithmetic. Free modules and their resolutions became central with the rise of homological algebra in the mid-twentieth century, where they measure how far modules deviate from being free.

Key figures

• Emmy Noether
• Heinrich Brandt
• Wolfgang Krull

Seminal works

• dummit2004
• lang2002
• atiyah1969

Frequently asked questions

Is every module free?

No. Over a field every module is free, but over a general ring most modules are not; for instance the integers modulo n has no basis as a module over the integers. Free modules are precisely those that do admit a basis.

How do projective modules relate to free modules?

Projective modules are exactly the direct summands of free modules, a slightly larger class. Over some rings, such as principal-ideal domains, finitely generated projective and free modules coincide, but in general they differ.