# Projective module

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Direct summand of a free module (mathematics)

In [mathematics](/source/Mathematics), particularly in [algebra](/source/Algebra), the [class](/source/Class_(set_theory)) of **projective modules** enlarges the class of [free modules](/source/Free_module) (that is, [modules](/source/Module_(mathematics)) with [basis vectors](/source/Basis_vector)) over a [ring](/source/Ring_(mathematics)), keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

Every free module is a projective module, but the [converse](/source/Converse_(logic)) fails to hold over some rings, such as [Dedekind rings](/source/Dedekind_ring) that are not [principal ideal domains](/source/Principal_ideal_domain). However, every projective module is a free module if the ring is a principal ideal domain such as the [integers](/source/Integer), or a (multivariate) [polynomial ring](/source/Polynomial_ring) over a [field](/source/Field_(mathematics)) (this is the [Quillen–Suslin theorem](/source/Quillen%E2%80%93Suslin_theorem)).

Projective modules were first introduced in 1956 in the influential book *Homological Algebra* by [Henri Cartan](/source/Henri_Cartan) and [Samuel Eilenberg](/source/Samuel_Eilenberg).

## Definitions

### Lifting property

The usual [category theoretical](/source/Category_theory) definition is in terms of the property of [*lifting*](/source/Lifting_property) that carries over from free to projective modules: a module *P* is projective [if and only if](/source/If_and_only_if) for every [surjective](/source/Surjective) [module homomorphism](/source/Module_homomorphism) *f* : *N* ↠ *M* and every module homomorphism *g* : *P* → *M*, there exists a module homomorphism *h* : *P* → *N* such that *fh* = *g*. (We don't require the lifting homomorphism *h* to be unique; this is not a [universal property](/source/Universal_property).)

The advantage of this definition of "projective" is that it can be carried out in [categories](/source/Category_(mathematics)) more general than [module categories](/source/Module_categories): we don't need a notion of "free object". It can also be [dualized](/source/Dual_(category_theory)), leading to [injective modules](/source/Injective_module). The lifting property may also be rephrased as *every morphism from P {\displaystyle P} to M {\displaystyle M} factors through every epimorphism to M {\displaystyle M}*. Thus, by definition, projective modules are precisely the [projective objects](/source/Projective_object) in the [category of *R*-modules](/source/Category_of_modules).

### Split-exact sequences

A module *P* is projective if and only if every [short exact sequence](/source/Short_exact_sequence) of modules of the form

- 0 → A → B → P → 0 {\displaystyle 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0}

is a [split exact sequence](/source/Split_exact_sequence). That is, for every surjective module homomorphism *f* : *B* ↠ *P* there exists a **section map**, that is, a module homomorphism *h* : *P* → *B* such that *fh* = id*P*. In that case, *h*(*P*) is a [direct summand](/source/Direct_summand) of *B*, *h* is an [isomorphism](/source/Isomorphism) from *P* to *h*(*P*), and *hf* is a [projection](/source/Projection_(linear_algebra)) on the summand *h*(*P*). Equivalently,

- B = Im ⁡ ( h ) ⊕ Ker ⁡ ( f ) where Ker ⁡ ( f ) ≅ A and Im ⁡ ( h ) ≅ P . {\displaystyle B=\operatorname {Im} (h)\oplus \operatorname {Ker} (f)\ \ {\text{ where }}\operatorname {Ker} (f)\cong A\ {\text{ and }}\operatorname {Im} (h)\cong P.}

### Direct summands of free modules

A module *P* is projective if and only if there is another module *Q* such that the [direct sum](/source/Direct_sum_of_modules) of *P* and *Q* is a free module.

### Exactness

An *R*-module *P* is projective if and only if the covariant [functor](/source/Functor) Hom(*P*, -): *R*-**Mod** → **Ab** is an [exact functor](/source/Exact_functor), where *R*-**Mod** is the category of left *R*-modules and **Ab** is the [category of abelian groups](/source/Category_of_abelian_groups). When the ring *R* is [commutative](/source/Commutative_ring), **Ab** is advantageously replaced by *R*-**Mod** in the preceding characterization. This functor is always [left exact](/source/Left_exact_functor), but, when *P* is projective, it is also right exact. This means that *P* is projective if and only if this functor preserves [epimorphisms](/source/Epimorphism) (surjective homomorphisms), or if it preserves finite [colimits](/source/Colimit).

### Dual basis

A module *P* is projective if and only if there exists a set { a i ∈ P ∣ i ∈ I } {\displaystyle \{a_{i}\in P\mid i\in I\}} and a set { f i ∈ H o m ( P , R ) ∣ i ∈ I } {\displaystyle \{f_{i}\in \mathrm {Hom} (P,R)\mid i\in I\}} such that for every *x* in *P*, *f**i*(*x*) is only nonzero for finitely many *i*, and x = ∑ f i ( x ) a i {\displaystyle x=\sum f_{i}(x)a_{i}} .

## Elementary examples and properties

The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules:

- Direct sums and direct summands of projective modules are projective.

- If *e* = *e*2 is an [idempotent](/source/Idempotent_(ring_theory)) in the ring *R*, then *Re* is a projective left module over *R*.

Let R = R 1 × R 2 {\displaystyle R=R_{1}\times R_{2}} be the [direct product](/source/Direct_product) of two rings R 1 {\displaystyle R_{1}} and R 2 , {\displaystyle R_{2},} which is a ring with operations defined componentwise. Let e 1 = ( 1 , 0 ) {\displaystyle e_{1}=(1,0)} and e 2 = ( 0 , 1 ) . {\displaystyle e_{2}=(0,1).} Then e 1 {\displaystyle e_{1}} and e 2 {\displaystyle e_{2}} are idempotents, and belong to the [centre](/source/Centre_of_a_ring) of R . {\displaystyle R.} The [two-sided ideals](/source/Two-sided_ideal) R e 1 {\displaystyle Re_{1}} and R e 2 {\displaystyle Re_{2}} are projective modules, since their direct sum (as R-modules) equals the free R-module R. However, if R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} are nontrivial, then they are not free as modules over R {\displaystyle R} . For instance Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is projective but not free over Z / 6 Z {\displaystyle \mathbb {Z} /6\mathbb {Z} } .

## Relation to other module-theoretic properties

The relation of projective modules to free and [flat](/source/Flat_module) modules is subsumed in the following diagram of module properties:

The left-to-right implications are true over any ring, although some authors define [torsion-free modules](/source/Torsion-free_module) only over a [domain](/source/Domain_(ring_theory)). The right-to-left implications are true over the rings labeling them. There may be other rings over which they are true. For example, the implication labeled "[local ring](/source/Local_ring) or PID" is also true for (multivariate) polynomial rings over a [field](/source/Field_(mathematics)): this is the [Quillen–Suslin theorem](/source/Quillen%E2%80%93Suslin_theorem).

### Projective vs. free modules

Any free module is projective. The converse is true in the following cases:

- if *R* is a field or [skew field](/source/Skew_field): *any* module is free in this case.

- if the ring *R* is a [principal ideal domain](/source/Principal_ideal_domain). For example, this applies to *R* = **Z** (the [integers](/source/Integer)), so an [abelian group](/source/Abelian_group) is projective if and only if it is a [free abelian group](/source/Free_abelian_group). The reason is that any [submodule](/source/Submodule) of a free module over a principal ideal domain is free.

- if the ring *R* is a [local ring](/source/Local_ring). This fact is the basis of the intuition of "locally free = projective". This fact is easy to [prove](/source/Mathematical_proof) for [finitely generated](/source/Finitely_generated_module) projective modules. In general, it is due to [Kaplansky (1958)](#CITEREFKaplansky1958); see [Kaplansky's theorem on projective modules](/source/Kaplansky's_theorem_on_projective_modules).

In general though, projective modules need not be free:

- Over a [direct product of rings](/source/Direct_product_of_rings) *R* × *S* where *R* and *S* are [nonzero](/source/Zero_ring) rings, both *R* × 0 and 0 × *S* are non-free projective modules.

- Over a [Dedekind domain](/source/Dedekind_domain) a non-[principal](/source/Principal_ideal) [ideal](/source/Ideal_(ring_theory)) is always a projective module that is not a free module.

- Over a [matrix ring](/source/Matrix_ring) M*n*(*R*), the natural module *R**n* is projective but is not free when *n* > 1.

- Over a [semisimple ring](/source/Semisimple_ring), *every* module is projective, but a nonzero proper left (or right) ideal is not a free module. Thus the only semisimple rings for which all projectives are free are [division rings](/source/Division_ring).

The difference between free and projective modules is, in a sense, measured by the [algebraic *K*-theory](/source/Algebraic_K-theory) [group](/source/Group_(mathematics)) *K*0(*R*); see below.

### Projective vs. flat modules

Every projective module is [flat](/source/Flat_module).[1] The converse is in general not true: the abelian group **Q** is a **Z**-module that is flat, but not projective.[2]

Conversely, a [finitely related](/source/Finitely_related_module) flat module is projective.[3]

[Govorov (1965)](#CITEREFGovorov1965) and [Lazard (1969)](#CITEREFLazard1969) proved that a module *M* is flat if and only if it is a [direct limit](/source/Direct_limit) of [finitely-generated](/source/Finitely_generated_module) [free modules](/source/Free_module).

In general, the precise relation between flatness and projectivity was established by [Raynaud & Gruson (1971)](#CITEREFRaynaudGruson1971) (see also [Drinfeld (2006)](#CITEREFDrinfeld2006) and [Braunling, Groechenig & Wolfson (2016)](#CITEREFBraunlingGroechenigWolfson2016)) who showed that a module *M* is projective if and only if it satisfies the following conditions:

- *M* is flat,

- *M* is a direct sum of [countably](/source/Countable_set) generated modules,

- *M* satisfies a certain [Mittag-Leffler](/source/G%C3%B6sta_Mittag-Leffler)-type condition.

This characterization can be used to show that if R → S {\displaystyle R\to S} is a [faithfully flat](/source/Faithfully_flat_morphism) map of commutative rings and M {\displaystyle M} is an R {\displaystyle R} -module, then M {\displaystyle M} is projective if and only if M ⊗ R S {\displaystyle M\otimes _{R}S} is projective.[4] In other words, the property of being projective satisfies [faithfully flat descent](/source/Faithfully_flat_descent).

## The category of projective modules

Submodules of projective modules need not be projective; a ring *R* for which every submodule of a projective left module is projective is called [left hereditary](/source/Hereditary_ring).

[Quotients](/source/Quotient_module) of projective modules also need not be projective, for example **Z**/*n* is a quotient of **Z**, but not [torsion-free](/source/Torsion-free_module), hence not flat, and therefore not projective.

The category of finitely generated projective modules over a ring is an [exact category](/source/Exact_category). (See also [algebraic K-theory](/source/Algebraic_K-theory)).

## Projective resolutions

Main article: [Projective resolution](/source/Projective_resolution)

Given a module, *M*, a **projective [resolution](/source/Resolution_(algebra))** of *M* is an infinite [exact sequence](/source/Exact_sequence) of modules

- ⋅⋅⋅ → *P**n* → ⋅⋅⋅ → *P*2 → *P*1 → *P*0 → *M* → 0,

with all the *P**i* s projective. Every module possesses a projective resolution. In fact a **free resolution** (resolution by free modules) exists. The exact sequence of projective modules may sometimes be abbreviated to *P*(*M*) → *M* → 0 or *P*• → *M* → 0. A classic example of a projective resolution is given by the [Koszul complex](/source/Koszul_complex) of a [regular sequence](/source/Regular_sequence), which is a free resolution of the [ideal](/source/Ideal_(ring_theory)) generated by the sequence.

The *length* of a finite resolution is the index *n* such that *P**n* is [nonzero](/source/Zero_module) and *P**i* = 0 for *i* greater than *n*. If *M* admits a finite projective resolution, the minimal length among all finite projective resolutions of *M* is called its **projective dimension** and denoted pd(*M*). If *M* does not admit a finite projective resolution, then by convention the projective dimension is said to be infinite. As an example, consider a module *M* such that pd(*M*) = 0. In this situation, the exactness of the sequence 0 → *P*0 → *M* → 0 indicates that the arrow in the center is an isomorphism, and hence *M* itself is projective.

## Projective modules over commutative rings

Projective modules over [commutative rings](/source/Commutative_ring) have nice properties.

The [localization](/source/Localization_(commutative_algebra)) of a projective module is a projective module over the localized ring. A projective module over a [local ring](/source/Local_ring) is free. Thus a projective module is *locally free* (in the sense that its localization at every [prime ideal](/source/Prime_ideal) is free over the corresponding localization of the ring). The converse is true for [finitely generated modules](/source/Finitely_generated_module) over [Noetherian rings](/source/Noetherian_ring): a finitely generated module over a commutative Noetherian ring is locally free if and only if it is projective.

However, there are examples of finitely generated modules over a non-Noetherian ring that are locally free and not projective. For instance, a [Boolean ring](/source/Boolean_ring) has all of its localizations isomorphic to **F**2, the field of two elements, so any module over a Boolean ring is locally free, but there are some non-projective modules over Boolean rings. One example is *R*/*I* where *R* is a direct product of countably many copies of **F**2 and *I* is the direct sum of countably many copies of **F**2 inside of *R*. The *R*-module *R*/*I* is locally free since *R* is Boolean (and it is finitely generated as an *R*-module too, with a spanning set of size 1), but *R*/*I* is not projective because *I* is not a principal ideal. (If a quotient module *R*/*I*, for any commutative ring *R* and ideal *I*, is a projective *R*-module then *I* is principal.)

However, it is true that for [finitely presented modules](/source/Finitely_presented_module) *M* over a commutative ring *R* (in particular if *M* is a finitely generated *R*-module and *R* is Noetherian), the following are equivalent.[5]

1. M {\displaystyle M} is flat.

1. M {\displaystyle M} is projective.

1. M m {\displaystyle M_{\mathfrak {m}}} is free as R m {\displaystyle R_{\mathfrak {m}}} -module for every [maximal ideal](/source/Maximal_ideal) m {\displaystyle {\mathfrak {m}}} of *R*.

1. M p {\displaystyle M_{\mathfrak {p}}} is free as R p {\displaystyle R_{\mathfrak {p}}} -module for every prime ideal p {\displaystyle {\mathfrak {p}}} of *R*.

1. There exist f 1 , … , f n ∈ R {\displaystyle f_{1},\ldots ,f_{n}\in R} generating the [unit ideal](/source/Unit_ideal) such that M [ f i − 1 ] {\displaystyle M[f_{i}^{-1}]} is free as R [ f i − 1 ] {\displaystyle R[f_{i}^{-1}]} -module for each *i*.

1. M ~ {\displaystyle {\widetilde {M}}} is a [locally free sheaf](/source/Locally_free_sheaf) on the [affine scheme](/source/Affine_scheme) Spec ⁡ R {\displaystyle \operatorname {Spec} R} (where M ~ {\displaystyle {\widetilde {M}}} is the [sheaf associated to](/source/Sheaf_associated_to_a_module) *M*.)

Moreover, if *R* is a Noetherian [integral domain](/source/Integral_domain), then, by [Nakayama's lemma](/source/Nakayama's_lemma), these conditions are equivalent to

- The [dimension](/source/Dimension_(vector_space)) of the k ( p ) {\displaystyle k({\mathfrak {p}})} -[vector space](/source/Vector_space) M ⊗ R k ( p ) {\displaystyle M\otimes _{R}k({\mathfrak {p}})} is the same for all prime ideals p {\displaystyle {\mathfrak {p}}} of *R,* where k ( p ) {\displaystyle k({\mathfrak {p}})} is the residue field at p {\displaystyle {\mathfrak {p}}} .[6] That is to say, *M* has constant rank (as defined below).

Let *A* be a commutative ring. If *B* is a (possibly non-commutative) *A*-[algebra](/source/Algebra_over_a_ring) that is a finitely generated projective *A*-module containing *A* as a [subring](/source/Subring), then *A* is a direct factor of *B*.[7]

### Rank

Let *P* be a finitely generated projective module over a commutative ring *R* and *X* be the [spectrum](/source/Spectrum_of_a_ring) of *R*. The *rank* of *P* at a prime ideal p {\displaystyle {\mathfrak {p}}} in *X* is the rank of the free R p {\displaystyle R_{\mathfrak {p}}} -module P p {\displaystyle P_{\mathfrak {p}}} . It is a locally constant function on *X*. In particular, if *X* is connected (that is if *R* has no other idempotents than 0 and 1), then *P* has constant rank.

## Vector bundles and locally free modules

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A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of [vector bundles](/source/Vector_bundle). This can be made precise for the ring of [continuous](/source/Continuous_function_(topology)) [real](/source/Real_number)-valued functions on a [compact](/source/Compact_space) [Hausdorff space](/source/Hausdorff_space), as well as for the ring of [smooth functions](/source/Smooth_function) on a [smooth manifold](/source/Manifold) (see [Serre–Swan theorem](/source/Serre%E2%80%93Swan_theorem) that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a [smooth vector bundle](/source/Smooth_vector_bundle)).

Vector bundles are *locally free*. If there is some notion of "localization" that can be carried over to modules, such as the usual [localization of a ring](/source/Localization_of_a_ring), one can define locally free modules, and the projective modules then typically coincide with the locally free modules.

## Projective modules over a polynomial ring

The [Quillen–Suslin theorem](/source/Quillen%E2%80%93Suslin_theorem), which solves Serre's problem, is another [deep result](/source/Deep_result): if *K* is a field, or more generally a [principal ideal domain](/source/Principal_ideal_domain), and *R* = *K*[*X*1,...,*X**n*] is a [polynomial ring](/source/Polynomial_ring) over *K*, then every projective module over *R* is free. This problem was first raised by Serre with *K* a field (and the modules being finitely generated). [Bass](/source/Hyman_Bass) settled it for non-finitely generated modules,[8] and [Quillen](/source/Dan_Quillen) and [Suslin](/source/Andrei_Suslin) independently and simultaneously treated the case of finitely generated modules.

Since every projective module over a principal ideal domain is free, one might ask this question: if *R* is a commutative ring such that every (finitely generated) projective *R*-module is free, then is every (finitely generated) projective *R*[*X*]-module free? The answer is *no*. A [counterexample](/source/Counterexample) occurs with *R* equal to the local ring of the curve *y*2 = *x*3 at the origin. Thus the Quillen–Suslin theorem could never be proved by a simple [induction](/source/Mathematical_induction) on the number of variables.

## See also

The Wikibook *[Commutative Algebra](https://en.wikibooks.org/wiki/Commutative_Algebra)* has a page on the topic of: ***[Torsion-free, flat, projective and free modules](https://en.wikibooks.org/wiki/Commutative_Algebra/Torsion-free,_flat,_projective_and_free_modules)***

- [Projective cover](/source/Projective_cover)

- [Schanuel's lemma](/source/Schanuel's_lemma)

- [Bass cancellation theorem](/source/Bass_cancellation_theorem)

- [Modular representation theory](/source/Modular_representation_theory)

## Notes

1. **[^](#cite_ref-1)** Hazewinkel; et al. (2004). "Corollary 5.4.5". [*Algebras, Rings and Modules, Part 1*](https://books.google.com/books?id=AibpdVNkFDYC&pg=PA131&dq=%22Every+projective+module+is+flat%22). p. 131.

1. **[^](#cite_ref-2)** Hazewinkel; et al. (2004). "Remark after Corollary 5.4.5". [*Algebras, Rings and Modules, Part 1*](https://books.google.com/books?id=AibpdVNkFDYC&pg=PA132&dq=%22Q+is+flat+but+it+is+not+projective%22). pp. 131–132.

1. **[^](#cite_ref-3)** [Cohn 2003](#CITEREFCohn2003), Corollary 4.6.4 harvnb error: no target: CITEREFCohn2003 ([help](https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors))

1. **[^](#cite_ref-4)** ["Section 10.95 (05A4): Descending properties of modules—The Stacks project"](https://stacks.math.columbia.edu/tag/05A4). *stacks.math.columbia.edu*. Retrieved 2022-11-03.

1. **[^](#cite_ref-5)** Exercises 4.11 and 4.12 and Corollary 6.6 of David Eisenbud, *Commutative Algebra with a view towards Algebraic Geometry*, GTM 150, Springer-Verlag, 1995. Also, [Milne 1980](#CITEREFMilne1980)

1. **[^](#cite_ref-6)** That is, k ( p ) = R p / p R p {\displaystyle k({\mathfrak {p}})=R_{\mathfrak {p}}/{\mathfrak {p}}R_{\mathfrak {p}}} is the residue field of the local ring R p {\displaystyle R_{\mathfrak {p}}} .

1. **[^](#cite_ref-7)** [Bourbaki, Algèbre commutative 1989](#CITEREFBourbaki,_Algèbre_commutative1989), Ch II, §5, Exercise 4 harvnb error: no target: CITEREFBourbaki,_Algèbre_commutative1989 ([help](https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors))

1. **[^](#cite_ref-8)** Bass, Hyman (1963). ["Big projective modules are free"](https://doi.org/10.1215%2Fijm%2F1255637479). *[Illinois Journal of Mathematics](/source/Illinois_Journal_of_Mathematics)*. **7** (1). Duke University Press. Corollary 4.5. [doi](/source/Doi_(identifier)):[10.1215/ijm/1255637479](https://doi.org/10.1215%2Fijm%2F1255637479).

## References

- William A. Adkins; Steven H. Weintraub (1992). [*Algebra: An Approach via Module Theory*](https://archive.org/details/springer_10.1007-978-1-4612-0923-2). Springer. Sec 3.5. [ISBN](/source/ISBN_(identifier)) [978-1-4612-0923-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4612-0923-2).

- Iain T. Adamson (1972). *Elementary rings and modules*. University Mathematical Texts. Oliver and Boyd. [ISBN](/source/ISBN_(identifier)) [0-05-002192-3](https://en.wikipedia.org/wiki/Special:BookSources/0-05-002192-3).

- [Nicolas Bourbaki](/source/Nicolas_Bourbaki), Commutative algebra, Ch. II, §5

- Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse (2016). "Tate Objects in Exact Categories (With an appendix by Jan Stovicek and Jan Trlifaj)". *Moscow Mathematical Journal*. **16** (3): 433–504. [arXiv](/source/ArXiv_(identifier)):[1402.4969v4](https://arxiv.org/abs/1402.4969v4). [doi](/source/Doi_(identifier)):[10.17323/1609-4514-2016-16-3-433-504](https://doi.org/10.17323%2F1609-4514-2016-16-3-433-504). [MR](/source/MR_(identifier)) [3510209](https://mathscinet.ams.org/mathscinet-getitem?mr=3510209). [S2CID](/source/S2CID_(identifier)) [118374422](https://api.semanticscholar.org/CorpusID:118374422).

- [Paul M. Cohn](/source/Paul_Cohn) (2003). *Further algebra and applications*. Springer. [ISBN](/source/ISBN_(identifier)) [1-85233-667-6](https://en.wikipedia.org/wiki/Special:BookSources/1-85233-667-6).

- [Drinfeld, Vladimir](/source/Vladimir_Drinfeld) (2006). "Infinite-dimensional vector bundles in algebraic geometry: an introduction". In [Pavel Etingof](/source/Pavel_Etingof); Vladimir Retakh; [I. M. Singer](/source/Isadore_Singer) (eds.). *The Unity of Mathematics*. Birkhäuser Boston. pp. 263–304. [arXiv](/source/ArXiv_(identifier)):[math/0309155v4](https://arxiv.org/abs/math/0309155v4). [doi](/source/Doi_(identifier)):[10.1007/0-8176-4467-9_7](https://doi.org/10.1007%2F0-8176-4467-9_7). [ISBN](/source/ISBN_(identifier)) [978-0-8176-4076-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8176-4076-7). [MR](/source/MR_(identifier)) [2181808](https://mathscinet.ams.org/mathscinet-getitem?mr=2181808).

- Govorov, V. E. (1965). "On flat modules (Russian)". *[Siberian Math. J.](/source/Siberian_Math._J.)* **6**: 300–304.

- [Hazewinkel, Michiel](/source/Michiel_Hazewinkel); [Gubareni, Nadiya](https://en.wikipedia.org/w/index.php?title=Nadiya_Gubareni&action=edit&redlink=1); [Kirichenko, Vladimir V.](https://en.wikipedia.org/w/index.php?title=Vladimir_V._Kirichenko&action=edit&redlink=1) (2004). *Algebras, rings and modules*. [Springer Science](/source/Springer_Science). [ISBN](/source/ISBN_(identifier)) [978-1-4020-2690-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-2690-4).

- [Kaplansky, Irving](/source/Irving_Kaplansky) (1958). "Projective modules". *[Ann. of Math.](/source/Ann._of_Math.)* 2. **68** (2): 372–377. [doi](/source/Doi_(identifier)):[10.2307/1970252](https://doi.org/10.2307%2F1970252). [hdl](/source/Hdl_(identifier)):[10338.dmlcz/101124](https://hdl.handle.net/10338.dmlcz%2F101124). [JSTOR](/source/JSTOR_(identifier)) [1970252](https://www.jstor.org/stable/1970252). [MR](/source/MR_(identifier)) [0100017](https://mathscinet.ams.org/mathscinet-getitem?mr=0100017).

- [Lang, Serge](/source/Serge_Lang) (1993). *Algebra* (3rd ed.). [Addison–Wesley](/source/Addison%E2%80%93Wesley). [ISBN](/source/ISBN_(identifier)) [0-201-55540-9](https://en.wikipedia.org/wiki/Special:BookSources/0-201-55540-9).

- [Lazard, D.](/source/Daniel_Lazard) (1969). ["Autour de la platitude"](https://doi.org/10.24033%2Fbsmf.1675). *[Bulletin de la Société Mathématique de France](/source/Bulletin_de_la_Soci%C3%A9t%C3%A9_Math%C3%A9matique_de_France)*. **97**: 81–128. [doi](/source/Doi_(identifier)):[10.24033/bsmf.1675](https://doi.org/10.24033%2Fbsmf.1675).

- Milne, James (1980). [*Étale cohomology*](https://archive.org/details/etalecohomology00miln). Princeton Univ. Press. [ISBN](/source/ISBN_(identifier)) [0-691-08238-3](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08238-3).

- Donald S. Passman (2004) *A Course in Ring Theory*, especially chapter 2 Projective modules, pp 13–22, AMS Chelsea, [ISBN](/source/ISBN_(identifier)) [0-8218-3680-3](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-3680-3) .

- [Raynaud, Michel](/source/Michel_Raynaud); Gruson, Laurent (1971). "Critères de platitude et de projectivité. Techniques de "platification" d'un module". *[Invent. Math.](/source/Invent._Math.)* **13**: 1–89. [Bibcode](/source/Bibcode_(identifier)):[1971InMat..13....1R](https://ui.adsabs.harvard.edu/abs/1971InMat..13....1R). [doi](/source/Doi_(identifier)):[10.1007/BF01390094](https://doi.org/10.1007%2FBF01390094). [MR](/source/MR_(identifier)) [0308104](https://mathscinet.ams.org/mathscinet-getitem?mr=0308104). [S2CID](/source/S2CID_(identifier)) [117528099](https://api.semanticscholar.org/CorpusID:117528099).

- [Paulo Ribenboim](/source/Paulo_Ribenboim) (1969) *Rings and Modules*, §1.6 Projective modules, pp 19–24, [Interscience Publishers](/source/Interscience_Publishers).

- [Charles Weibel](/source/Charles_Weibel), [The K-book: An introduction to algebraic K-theory](http://www.math.rutgers.edu/~weibel/Kbook.html)

## Further reading

- [https://mathoverflow.net/questions/272018/faithfully-flat-descent-of-projectivity-for-non-commutative-rings](https://mathoverflow.net/questions/272018/faithfully-flat-descent-of-projectivity-for-non-commutative-rings)

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Adapted from the Wikipedia article [Projective module](https://en.wikipedia.org/wiki/Projective_module) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Projective_module?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.