Definition:Free Abelian Group on Set

Definition

Let $\Z$ be the additive group of integers.

Let $S$ be a set.

The free abelian group on $S$ is the pair $(\Z^{(S)}, \iota)$ where:

Also denoted as

The free abelian group on $S$ is also denoted $\Z[S]$. Not to be confused with a polynomial ring.