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:
 * $\Z^{(S)}$ is the direct sum of $S$ copies of $\Z$. That is, of the indexed family $S \to \{\Z\}$
 * $\iota : S \to \Z^{(S)}$ is the canonical mapping, which sends $s$ to the mapping $\delta_{st} \in \Z^{(S)}$, where $\delta$ denotes Kronecker delta.

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

Also see

 * Universal Property of Free Abelian Group on Set
 * Abelianization of Free Group is Free Abelian Group

Generalizations

 * Definition:Free Module on Set