Pure submodule

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, 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 of projective modules, and a pure exact sequence is a direct limit of split exact sequences.

Let R {\displaystyle R} be a ring (associative, with 1 {\displaystyle 1} ), let M {\displaystyle M} be a (left) module over R {\displaystyle R} , and let P {\displaystyle P} be a submodule of M {\displaystyle M} with ι : P ↪ M {\displaystyle \iota \colon P\hookrightarrow M} be the natural 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

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} .

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 ( 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 (also known as direct limit) of split exact sequences

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

• Every direct summand of M is pure in M. Consequently, every subspace of a vector space over a field is pure.

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 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, 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]
2. 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.
3. 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 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} .

• Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226

• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294