# Stably free module

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In [mathematics](/source/Mathematics), a **stably free module** is a [module](/source/Module_(mathematics)) which is close to being [free](/source/Free_module).

## Definition

A module *M* over a ring *R* is *stably free* if there exists a free [finitely generated module](/source/Finitely_generated_module) *F* over *R* such that M ⊕ F {\displaystyle M\oplus F} is a free module.

## Properties

- A [projective module](/source/Projective_module) is stably free if and only if it possesses a finite [free resolution](/source/Free_resolution).[1]

- An [infinitely generated module](https://en.wikipedia.org/w/index.php?title=Infinitely_generated_module&action=edit&redlink=1) is stably free if and only if it is free.[2]

## See also

- [Free object](/source/Free_object)

- [Eilenberg–Mazur swindle](/source/Eilenberg%E2%80%93Mazur_swindle)

- [Hermite ring](/source/Hermite_ring)

## References

1. **[^](#cite_ref-1)** [Lang, Serge](/source/Serge_Lang) (1993), *[Algebra](/source/Algebra_(Lang))* (Third ed.), Reading, Mass.: Addison-Wesley, [ISBN](/source/ISBN_(identifier)) [978-0-201-55540-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-55540-0), [Zbl](/source/Zbl_(identifier)) [0848.13001](https://zbmath.org/?format=complete&q=an:0848.13001)

1. **[^](#cite_ref-2)** [Lam, T. Y.](/source/Tsit_Yuen_Lam) (1978). [*Serre's Conjecture*](https://books.google.com/books?id=f-t6CwAAQBAJ&pg=PA23&dq=%22If+P+is+stably+free%2C+but+not+finitely+generated%2C+then+P+is+actually+free.%22). p. 23.

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