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

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

Generalizations

• Definition:Free Module on Set

Sources

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