# Pure submodule

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Module components with flexibility in module theory

In [mathematics](/source/Mathematics), especially in the field of [module theory](/source/Module_theory), the concept of **pure submodule** provides a generalization of [direct summand](/source/Direct_summand), a type of particularly well-behaved piece of a [module](/source/Module_(mathematics)). Pure modules are complementary to [flat modules](/source/Flat_module) and generalize Prüfer's notion of [pure subgroups](/source/Pure_subgroup). While flat modules are those modules which leave [short exact sequences](/source/Short_exact_sequence) exact after [tensoring](/source/Tensor_product), a pure submodule defines a short exact sequence (known as a **pure exact sequence**) that remains exact after tensoring with any module. Similarly a flat module is a [direct limit](/source/Direct_limit) of [projective modules](/source/Projective_module), and a pure exact sequence is a direct limit of [split exact sequences](/source/Split_exact_sequence).

## Definition

Let R {\displaystyle R} be a [ring](/source/Ring_(mathematics)) (associative, with 1 {\displaystyle 1} ), let M {\displaystyle M} be a (left) [module](/source/Module_(mathematics)) over R {\displaystyle R} , and let P {\displaystyle P} be a [submodule](/source/Submodule) of M {\displaystyle M} with ι : P ↪ M {\displaystyle \iota \colon P\hookrightarrow M} be the natural [injective](/source/Injective) map. Then P {\displaystyle P} is a **pure submodule of M {\displaystyle M}** if, for any (right) R {\displaystyle R} -module X {\displaystyle X} , the natural induced map i d X ⊗ R ι : X ⊗ R P → X ⊗ R M {\displaystyle \mathrm {id} _{X}\otimes _{R}\iota \colon X\otimes _{R}P\to X\otimes _{R}M} is injective.

Analogously, a [short exact sequence](/source/Short_exact_sequence)

- 0 ⟶ A ⟶ f B ⟶ g C ⟶ 0 {\displaystyle 0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0}

of (left) R {\displaystyle R} -modules is **pure exact** if the sequence stays exact when tensored with any (right) R {\displaystyle R} -module X {\displaystyle X} . This is equivalent to saying that f ( A ) {\displaystyle f(A)} is a pure submodule of B {\displaystyle B} .

## Equivalent characterizations

Purity of a submodule can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, P {\displaystyle P} is pure in M {\displaystyle M} if and only if the following condition holds: for any m {\displaystyle m} -by- n {\displaystyle n} [matrix](/source/Matrix_(mathematics)) ( a i j ) {\displaystyle (a_{ij})} with entries in R {\displaystyle R} , and any set y 1 , ⋯ , y m {\displaystyle y_{1},\cdots ,y_{m}} of elements of P {\displaystyle P} , if there exist elements x 1 , ⋯ , x n {\displaystyle x_{1},\cdots ,x_{n}} **in M {\displaystyle M}** such that

- ∑ j = 1 n a i j x j = y i for i = 1 , … , m {\displaystyle \sum _{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m}

then there also exist elements x 1 ′ , ⋯ , x n ′ {\displaystyle x'_{1},\cdots ,x'_{n}} **in P {\displaystyle P}** such that

- ∑ j = 1 n a i j x j ′ = y i for i = 1 , … , m {\displaystyle \sum _{j=1}^{n}a_{ij}x'_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m}

Another characterization is: a sequence is pure exact if and only if it is the [filtered colimit](/source/Filtered_colimit) (also known as [direct limit](/source/Direct_limit)) of [split exact sequences](/source/Split_exact_sequence)

- 0 ⟶ A i ⟶ B i ⟶ C i ⟶ 0. {\displaystyle 0\longrightarrow A_{i}\longrightarrow B_{i}\longrightarrow C_{i}\longrightarrow 0.} [1]

## Examples

- Every [direct summand](/source/Direct_summand) of *M* is pure in *M*. Consequently, every [subspace](/source/Linear_subspace) of a [vector space](/source/Vector_space) over a [field](/source/Field_(mathematics)) is pure.

## Properties

Suppose 0 ⟶ A ⟶ f B ⟶ g C ⟶ 0 {\displaystyle 0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0} is a short exact sequence of R {\displaystyle R} -modules, then:

1. C {\displaystyle C} is a [flat module](/source/Flat_module) if and only if the exact sequence is pure exact for every A {\displaystyle A} and B {\displaystyle B} . From this we can deduce that over a [von Neumann regular ring](/source/Von_Neumann_regular_ring), *every* submodule of *every* R {\displaystyle R} -module is pure. This is because *every* module over a von Neumann regular ring is flat. The converse is also true.[2]

1. Suppose B {\displaystyle B} is flat. Then the sequence is pure exact if and only if C {\displaystyle C} is flat. From this one can deduce that pure submodules of flat modules are flat.

1. Suppose C {\displaystyle C} is flat. Then B {\displaystyle B} is flat if and only if A {\displaystyle A} is flat.

If 0 ⟶ A ⟶ f B ⟶ g C ⟶ 0 {\displaystyle 0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0} is pure-exact, and F {\displaystyle F} is a [finitely presented](/source/Finitely_presented_module) R {\displaystyle R} -module, then every homomorphism from F {\displaystyle F} to C {\displaystyle C} can be lifted to B {\displaystyle B} , i.e. to every u : F → C {\displaystyle u\colon F\to C} there exists v : F → B {\displaystyle v\colon F\to B} such that g v = u {\displaystyle gv=u} .

## References

1. **[^](#cite_ref-1)** For abelian groups, this is proved in [Fuchs (2015](#CITEREFFuchs2015), Ch. 5, Thm. 3.4)

1. **[^](#cite_ref-FOOTNOTELam1999162_2-0)** [Lam 1999](#CITEREFLam1999), p. 162.

- Fuchs, László (2015), *Abelian Groups*, Springer Monographs in Mathematics, Springer, [ISBN](/source/ISBN_(identifier)) [9783319194226](https://en.wikipedia.org/wiki/Special:BookSources/9783319194226)

- [Lam, Tsit-Yuen](/source/Tsit_Yuen_Lam) (1999), *Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-98428-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98428-5), [MR](/source/MR_(identifier)) [1653294](https://mathscinet.ams.org/mathscinet-getitem?mr=1653294)

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