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 $H<_{p}G$ 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 $K<_{p}H$ , $H<_{p}G$ , 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 not pure.
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 over commutative 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 after tensoring 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 exact sequence

$0\rightarrow N\rightarrow M\rightarrow M/N\rightarrow 0$

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 $R$ 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	2013-03-22 14:57:47
Last modified on	2013-03-22 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