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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the field of module theory, the concept of pure submodule provides a generalization of
direct summand The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
, a type of particularly well-behaved piece of a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
. Pure modules are complementary to flat modules and generalize Prüfer's notion of
pure subgroup In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. Definition A subgroup S of a (typica ...
s. 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 In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of
projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
s, and a pure exact sequence is a direct limit of
split exact sequence In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. Equivalent characterizations A short exact sequence of abelian groups or of modules over a ...
s.


Definition

Let ''R'' be a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
(associative, with 1), let ''M'' be a (left)
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
over ''R'', let ''P'' be a submodule of ''M'' and let ''i'': ''P'' → ''M'' be the natural
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
map. Then ''P'' is a pure submodule of ''M'' if, for any (right) ''R''-module ''X'', the natural induced map id''X'' ⊗ ''i'' : ''X'' ⊗ ''P'' → ''X'' ⊗ ''M'' (where the on
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
s are taken over ''R'') is injective. Analogously, a short exact sequence :0 \longrightarrow A\,\ \stackrel\ B\,\ \stackrel\ 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''.


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'' is pure in ''M'' if and only if the following condition holds: for any ''m''-by-''n''
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(''a''''ij'') with entries in ''R'', and any set ''y''1, ..., ''y''''m'' of elements of ''P'', if there exist elements ''x''1, ..., ''x''''n'' in ''M'' such that :\sum_^n a_x_j = y_i \qquad\mbox i=1,\ldots,m then there also exist elements ''x''1′, ..., ''x''''n''′ in ''P'' such that :\sum_^n a_x'_j = y_i \qquad\mbox i=1,\ldots,m Another characterization is: a sequence is pure exact if and only if it is the
filtered colimit In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered c ...
(also known as
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
) of
split exact sequence In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. Equivalent characterizations A short exact sequence of abelian groups or of modules over a ...
s :0 \longrightarrow A_i \longrightarrow B_i \longrightarrow C_i \longrightarrow 0.For abelian groups, this is proved in


Examples

* Every
direct summand The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of ''M'' is pure in ''M''. Consequently, every subspace of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
is pure.


Properties

Suppose :0 \longrightarrow A\,\ \stackrel\ B\,\ \stackrel\ 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\ B\,\ \stackrel\ 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'' : ''F'' → ''C'' there exists ''v'' : ''F'' → ''B'' such that ''gv''=''u''.


References

* *{{Citation , last1=Lam , first1=Tsit-Yuen , title=Lectures on modules and rings , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, location=Berlin, New York , series=Graduate Texts in Mathematics No. 189 , isbn=978-0-387-98428-5 , mr=1653294 , year=1999 Module theory