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''in the sense of Cohn''
 
''in the sense of Cohn''
  
A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759101.png" /> of a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759102.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759103.png" /> such that for any left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759104.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759105.png" /> the natural homomorphism of Abelian groups
+
A submodule $  A $
 +
of a right $  R $-
 +
module $  B $
 +
such that for any left $  R $-
 +
module $  C $
 +
the natural homomorphism of Abelian groups
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759106.png" /></td> </tr></table>
+
$$
 +
A \otimes _ {R} C  \rightarrow  B \otimes _ {R} C
 +
$$
  
 
is injective. This is equivalent to the following condition: If the system of equations
 
is injective. This is equivalent to the following condition: If the system of equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759107.png" /></td> </tr></table>
+
$$
 
+
\sum_{i=1} ^ { n }  x _ {i} \lambda _ {ij}  = a _ {j} ,\ \
has a solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759108.png" />, then it has a solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p0759109.png" /> (cf. [[Flat module|Flat module]]). Any direct summand is a pure submodule. All submodules of a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591010.png" />-module are pure if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591011.png" /> is a [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]].
+
1 \leq j \leq m ,\ \
 
+
\lambda _ {ij} \in R ,\ a _ {j} \in A ,
In the case of Abelian groups (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591012.png" />), the following assertions are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591013.png" /> is a pure (or serving) subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591014.png" /> (cf. [[Pure subgroup|Pure subgroup]]); 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591015.png" /> for every natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591016.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591017.png" /> is a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591018.png" /> for every natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591019.png" />; 4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591021.png" /> is a finitely-generated group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591022.png" /> is a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591023.png" />; 5) every residue class in the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591024.png" /> contains an element of the same order as the residue class; and 6) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591026.png" /> is finitely generated, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591027.png" /> is a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591028.png" />. If property 2) is required to hold only for prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591030.png" /> is called a weakly-pure subgroup.
+
$$
 
 
The axiomatic approach to the notion of purity is based on the consideration of a class of monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591031.png" /> subject to the following conditions (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591032.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591033.png" /> is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591034.png" /> and that the natural imbedding belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591035.png" />): P0') if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591036.png" /> is a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591038.png" />; P1') if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591041.png" />; P2') if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591044.png" />; P3') if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591047.png" />; and P4') if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591051.png" />. Taking the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591052.png" /> instead of the class of all monomorphisms leads to [[Relative homological algebra|relative homological algebra]]. For example, a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591053.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591055.png" />-injective if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591056.png" /> implies that any homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591057.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591058.png" /> can be extended to a homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591059.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591060.png" /> (cf. [[Injective module|Injective module]]). A pure injective Abelian group is called algebraically compact. The following conditions on an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591061.png" /> are equivalent: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591062.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591063.png" /> is algebraically compact; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591064.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591065.png" /> splits as a direct summand of any group that contains it as a pure subgroup; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591066.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591067.png" /> is a direct summand of a group that admits a compact topology; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591068.png" />) a system of equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075910/p07591069.png" /> is solvable if every finite subsystem of it is solvable.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.P. Mishina,  L.A. Skornyakov,  "Abelian groups and modules" , Amer. Math. Soc. (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.G. Sklyarenko,   "Relative homological algebra in categories of modules"  ''Russian Math. Surveys'' , '''33''' : 3  (1978)  pp. 97–137  ''Uspekhi Mat. Nauk'' , '''33''' :  3  (1978)  pp. 85–120</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Fuchs,  "Infinite abelian groups" , '''1–2''' , Acad. Press  (1970–1973)</TD></TR></table>
 
 
 
  
 +
has a solution in  $  B $,
 +
then it has a solution in  $  A $(
 +
cf. [[Flat module|Flat module]]). Any direct summand is a pure submodule. All submodules of a right  $  R $-
 +
module are pure if and only if  $  R $
 +
is a [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]].
  
====Comments====
+
In the case of Abelian groups (that is,  $  R = \mathbf Z $),
 +
the following assertions are equivalent: 1)  $  A $
 +
is a pure (or serving) subgroup of  $  B $(
 +
cf. [[Pure subgroup|Pure subgroup]]); 2)  $  n A = A \cap n B $
 +
for every natural number  $  n $;
 +
3)  $  A / n A $
 +
is a direct summand of  $  B / n A $
 +
for every natural number  $  n $;
 +
4) if  $  C \subseteq A $
 +
and  $  A / C $
 +
is a finitely-generated group, then  $  A/C $
 +
is a direct summand of  $  B/C $;
 +
5) every residue class in the quotient group  $  B / A $
 +
contains an element of the same order as the residue class; and 6) if  $  A \subseteq C \subseteq B $
 +
and  $  C / A $
 +
is finitely generated, then  $  A $
 +
is a direct summand of  $  C $.
 +
If property 2) is required to hold only for prime numbers  $  n $,
 +
then  $  A $
 +
is called a weakly-pure subgroup.
  
 +
The axiomatic approach to the notion of purity is based on the consideration of a class of monomorphism  $  \mathfrak K _  \omega  $
 +
subject to the following conditions (here  $  A \subseteq _  \omega  B $
 +
means that  $  A $
 +
is a submodule of  $  B $
 +
and that the natural imbedding belongs to  $  \mathfrak K _  \omega  $):
 +
P0') if  $  A $
 +
is a direct summand of  $  B $,
 +
then  $  A \subseteq _  \omega  B $;
 +
P1') if  $  A \subseteq _  \omega  B $
 +
and  $  B \subseteq _  \omega  C $,
 +
then  $  A \subseteq _  \omega  C $;
 +
P2') if  $  A \subseteq B \subseteq C $
 +
and  $  A \subseteq _  \omega  C $,
 +
then  $  A \subseteq _  \omega  B $;
 +
P3') if  $  A \subseteq _  \omega  B $
 +
and  $  K \subseteq A $,
 +
then  $  A / K \subseteq _  \omega  B / K $;
 +
and P4') if  $  K \subseteq B $,
 +
$  K \subseteq _  \omega  B $
 +
and  $  A / K \subseteq _  \omega  B / K $,
 +
then  $  A \subseteq _  \omega  B $.
 +
Taking the class  $  \mathfrak K _  \omega  $
 +
instead of the class of all monomorphisms leads to [[Relative homological algebra|relative homological algebra]]. For example, a module  $  Q $
 +
is called  $  \omega $-
 +
injective if  $  A \subseteq _  \omega  B $
 +
implies that any homomorphism from  $  A $
 +
into  $  Q $
 +
can be extended to a homomorphism from  $  B $
 +
into  $  Q $(
 +
cf. [[Injective module|Injective module]]). A pure injective Abelian group is called algebraically compact. The following conditions on an Abelian group  $  Q $
 +
are equivalent:  $  \alpha $)
 +
$  Q $
 +
is algebraically compact;  $  \beta $)
 +
$  Q $
 +
splits as a direct summand of any group that contains it as a pure subgroup;  $  \gamma $)
 +
$  Q $
 +
is a direct summand of a group that admits a compact topology; and  $  \delta $)
 +
a system of equations over  $  Q $
 +
is solvable if every finite subsystem of it is solvable.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Rotman,  "Introduction to homological algebra" , Acad. Press  (1979)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Mishina,  L.A. Skornyakov,  "Abelian groups and modules" , Amer. Math. Soc.  (1976)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  E.G. Sklyarenko,  "Relative homological algebra in categories of modules"  ''Russian Math. Surveys'' , '''33''' :  3  (1978)  pp. 97–137  ''Uspekhi Mat. Nauk'' , '''33''' :  3  (1978)  pp. 85–120</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  L. Fuchs,  "Infinite abelian groups" , '''1–2''' , Acad. Press  (1970–1973)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Rotman,  "Introduction to homological algebra" , Acad. Press  (1979)</TD></TR>
 +
</table>

Latest revision as of 08:04, 21 January 2024


in the sense of Cohn

A submodule $ A $ of a right $ R $- module $ B $ such that for any left $ R $- module $ C $ the natural homomorphism of Abelian groups

$$ A \otimes _ {R} C \rightarrow B \otimes _ {R} C $$

is injective. This is equivalent to the following condition: If the system of equations

$$ \sum_{i=1} ^ { n } x _ {i} \lambda _ {ij} = a _ {j} ,\ \ 1 \leq j \leq m ,\ \ \lambda _ {ij} \in R ,\ a _ {j} \in A , $$

has a solution in $ B $, then it has a solution in $ A $( cf. Flat module). Any direct summand is a pure submodule. All submodules of a right $ R $- module are pure if and only if $ R $ is a regular ring (in the sense of von Neumann).

In the case of Abelian groups (that is, $ R = \mathbf Z $), the following assertions are equivalent: 1) $ A $ is a pure (or serving) subgroup of $ B $( cf. Pure subgroup); 2) $ n A = A \cap n B $ for every natural number $ n $; 3) $ A / n A $ is a direct summand of $ B / n A $ for every natural number $ n $; 4) if $ C \subseteq A $ and $ A / C $ is a finitely-generated group, then $ A/C $ is a direct summand of $ B/C $; 5) every residue class in the quotient group $ B / A $ contains an element of the same order as the residue class; and 6) if $ A \subseteq C \subseteq B $ and $ C / A $ is finitely generated, then $ A $ is a direct summand of $ C $. If property 2) is required to hold only for prime numbers $ n $, then $ A $ is called a weakly-pure subgroup.

The axiomatic approach to the notion of purity is based on the consideration of a class of monomorphism $ \mathfrak K _ \omega $ subject to the following conditions (here $ A \subseteq _ \omega B $ means that $ A $ is a submodule of $ B $ and that the natural imbedding belongs to $ \mathfrak K _ \omega $): P0') if $ A $ is a direct summand of $ B $, then $ A \subseteq _ \omega B $; P1') if $ A \subseteq _ \omega B $ and $ B \subseteq _ \omega C $, then $ A \subseteq _ \omega C $; P2') if $ A \subseteq B \subseteq C $ and $ A \subseteq _ \omega C $, then $ A \subseteq _ \omega B $; P3') if $ A \subseteq _ \omega B $ and $ K \subseteq A $, then $ A / K \subseteq _ \omega B / K $; and P4') if $ K \subseteq B $, $ K \subseteq _ \omega B $ and $ A / K \subseteq _ \omega B / K $, then $ A \subseteq _ \omega B $. Taking the class $ \mathfrak K _ \omega $ instead of the class of all monomorphisms leads to relative homological algebra. For example, a module $ Q $ is called $ \omega $- injective if $ A \subseteq _ \omega B $ implies that any homomorphism from $ A $ into $ Q $ can be extended to a homomorphism from $ B $ into $ Q $( cf. Injective module). A pure injective Abelian group is called algebraically compact. The following conditions on an Abelian group $ Q $ are equivalent: $ \alpha $) $ Q $ is algebraically compact; $ \beta $) $ Q $ splits as a direct summand of any group that contains it as a pure subgroup; $ \gamma $) $ Q $ is a direct summand of a group that admits a compact topology; and $ \delta $) a system of equations over $ Q $ is solvable if every finite subsystem of it is solvable.

References

[1] A.P. Mishina, L.A. Skornyakov, "Abelian groups and modules" , Amer. Math. Soc. (1976) (Translated from Russian)
[2] E.G. Sklyarenko, "Relative homological algebra in categories of modules" Russian Math. Surveys , 33 : 3 (1978) pp. 97–137 Uspekhi Mat. Nauk , 33 : 3 (1978) pp. 85–120
[3] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[4] L. Fuchs, "Infinite abelian groups" , 1–2 , Acad. Press (1970–1973)
[a1] J. Rotman, "Introduction to homological algebra" , Acad. Press (1979)
How to Cite This Entry:
Pure submodule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pure_submodule&oldid=18846
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article