# Definition:Free Abelian Group on Set

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## Definition

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

Let $S$ be a set.

The **free abelian group on $S$** is the pair $\struct {\Z^{\paren S}, \iota}$ where:

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

## Also denoted as

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

## Also see

### Generalizations

## Sources

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