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Pure submodule

Module components with flexibility in module theory

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Source updated April 22, 2026

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.

01Definition

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

Analogously, a short exact sequence

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

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

02Equivalent 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, $P$ is pure in $M$ if and only if the following condition holds: for any $m$ -by- $n$ matrix $(a_{ij})$ with entries in $R$ , and any set $y_{1},\cdots ,y_{m}$ of elements of $P$ , if there exist elements $x_{1},\cdots ,x_{n}$ in $M$ such that

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

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

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

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

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

03Examples

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

04Properties

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

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

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

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Sources and credits

This article is adapted from the Wikipedia article “Pure submodule”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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