# 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