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 $h=ma$ forsome integer $m$ and some $a\\in G$ , then there exists $b\\in H$ suchthat $h=mb$ . In other words, if $h$ is divisible in $G$ by aninteger, then it is divisible in $H$ by that same integer. Purity inabelian groups is a relative notion, and we denote $H<_{p}G$ to meanthat $H$ 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 trivialsubgroup 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 $K<_{p}H$ , $H<_{p}G$ , then $K<_{p}G$ .
5.
If $H=\\bigcup_{i=1}^{\\infty}H_{i}$ with $H_{i}\\leq H_{i+1}$ and $H_{i}<_{p}G$ , then $H<_{p}G$ .
6.
In $Z_{n^{2}}$ , $\\langle n\\rangle$ is an example of a subgroup that is notpure.
7.
In general, $\\langle m\\rangle<_{p}Z_{n}$ if $\\operatorname{gcd}(s,t)=1$ ,where $s=\\operatorname{gcd}(m,n)$ and $t=n/s$ .

Remark. This definition can be generalized to modules overcommutative rings.

Definition. Let $R$ be a commutative ring and $\\mathcal{E}\\colon 0\\rightarrow A\\rightarrow B\\rightarrow C\\rightarrow 0$ a short exact sequence of $R$ -modules. Then $\\mathcal{E}$ is said to be pure if it remains exact aftertensoring with any $R$ -module. In other words, if $D$ is any $R$ -module, then

D\\otimes\\mathcal{E}\\colon 0\\rightarrow D\\otimes A\\rightarrow D\\otimes B%\\rightarrow D\\otimes C\\rightarrow 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 exactsequence

0\\rightarrow N\\rightarrow M\\rightarrow M/N\\rightarrow 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 $R$ iffwhenever 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 aresult, if $I$ is an ideal of $R$ , then the purity of $N$ in $M$ meansthat $N\\cap IM=IN$ , which is a generalization of the secondcondition in the definition of a pure subgroup above.