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 ${\displaystyle R}$ be a ring (associative, with ${\displaystyle 1}$), let ${\displaystyle M}$ be a (left) module over ${\displaystyle R}$, and let ${\displaystyle P}$ be a submodule of ${\displaystyle M}$ with ${\displaystyle \iota \colon P\hookrightarrow M}$ be the natural injective map. Then ${\displaystyle P}$ is a pure submodule of ${\displaystyle M}$ if, for any (right) ${\displaystyle R}$-module ${\displaystyle X}$, the natural induced map ${\displaystyle \mathrm {id} _{X}\otimes _{R}\iota \colon X\otimes _{R}P\to X\otimes _{R}M}$ is injective.

Analogously, a short exact sequence

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

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

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, ${\displaystyle P}$ is pure in ${\displaystyle M}$ if and only if the following condition holds: for any ${\displaystyle m}$-by-${\displaystyle n}$ matrix ${\displaystyle (a_{ij})}$ with entries in ${\displaystyle R}$, and any set ${\displaystyle y_{1},\cdots ,y_{m}}$ of elements of ${\displaystyle P}$, if there exist elements ${\displaystyle x_{1},\cdots ,x_{n}}$ in ${\displaystyle M}$ such that

${\displaystyle \sum _{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m}$

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

${\displaystyle \sum _{j=1}^{n}a_{ij}x'_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m}$

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

${\displaystyle 0\longrightarrow A_{i}\longrightarrow B_{i}\longrightarrow C_{i}\longrightarrow 0.}$¹

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

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

1. ${\displaystyle C}$ is a flat module if and only if the exact sequence is pure exact for every ${\displaystyle A}$ and ${\displaystyle B}$. From this we can deduce that over a von Neumann regular ring, every submodule of every ${\displaystyle R}$-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true.²
2. Suppose ${\displaystyle B}$ is flat. Then the sequence is pure exact if and only if ${\displaystyle C}$ is flat. From this one can deduce that pure submodules of flat modules are flat.
3. Suppose ${\displaystyle C}$ is flat. Then ${\displaystyle B}$ is flat if and only if ${\displaystyle A}$ is flat.

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