pure subgroup
Definition. A pure subgroup $H$ of an abelian group^{} $G$ is

1.
a subgroup^{} of $G$, such that

2.
$H\cap mG=mH$ for all $m\in \mathbb{Z}$.
The second condition says that for any $h\in H$ such that $h=ma$ for some integer $m$ and some $a\in G$, then there exists $b\in H$ such that $h=mb$. In other words, if $h$ is divisible in $G$ by an integer, then it is divisible in $H$ by that same integer. Purity in abelian groups is a relative notion, and we denote $$ to mean that $H$ is a pure subgroup of $G$.
Examples. All groups mentioned below are abelian groups.

1.
For any group, two trivial examples of pure subgroups are the trivial subgroup and the group itself.

2.
Any divisible subgroup (http://planetmath.org/DivisibleGroup) or any direct summand^{} of a group is pure.

3.
The torsion subgroup (= the subgroup of all torsion elements) of any group is pure.

4.
If $$, $$, then $$.

5.
If $H={\bigcup}_{i=1}^{\mathrm{\infty}}{H}_{i}$ with ${H}_{i}\le {H}_{i+1}$ and $$, then $$.

6.
In ${Z}_{{n}^{2}}$, $\u27e8n\u27e9$ is an example of a subgroup that is not pure.

7.
In general, $$ if $\mathrm{gcd}(s,t)=1$, where $s=\mathrm{gcd}(m,n)$ and $t=n/s$.
Remark. This definition can be generalized to modules over commutative rings.
Definition. Let $R$ be a commutative ring and $\mathcal{E}:0\to A\to B\to C\to 0$ a short exact sequence^{} of $R$modules. Then $\mathcal{E}$ is said to be pure if it remains exact after tensoring with any $R$module. In other words, if $D$ is any $R$module, then
$$D\otimes \mathcal{E}:0\to D\otimes A\to D\otimes B\to D\otimes C\to 0,$$ 
is exact.
Definition. Let $N$ be a submodule^{} of $M$ over a ring $R$. Then $N$ is said to be a pure submodule of $M$ if the exact sequence^{}
$$0\to N\to M\to M/N\to 0$$ 
is a pure exact sequence.
From this definition, it is clear that $H$ is a pure subgroup of $G$ iff $H$ is a pure $\mathbb{Z}$submodule of $G$.
Remark. $N$ is a pure submodule of $M$ over $R$ iff whenever a finite sum
$$\sum {r}_{i}{m}_{i}=n\in N,$$ 
where ${m}_{i}\in M$ and ${r}_{i}\in R$ implies that
$$n=\sum {r}_{i}{n}_{i}$$ 
for some ${n}_{i}\in N$. As a result, if $I$ is an ideal of $R$, then the purity of $N$ in $M$ means that $N\cap IM=IN$, which is a generalization of the second condition in the definition of a pure subgroup above.
Title  pure subgroup 

Canonical name  PureSubgroup 
Date of creation  20130322 14:57:47 
Last modified on  20130322 14:57:47 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 20K27 
Classification  msc 13C13 
Defines  pure submodule 
Defines  pure exact sequence 