Rank of an abelian group

Remarks

Example: $A = \Z^2 \oplus \Z/3\Z$ has rank $2$. The set $\set{(1, 0, 0), (0, 1, 0)}$ is independent, and it is maximal: for any $y \in A$, $3y$ lies in $\Z^2 \oplus 0$, so $3y = m(1,0,0) + n(0,1,0)$ gives a relation with nonzero coefficient on $y$.

Non-example of independence: in $\Z/6\Z$, the set $\set{1}$ is not independent, since $6 \cdot 1 = 0$ with coefficient $6 \neq 0$. In fact a group in which every element has finite order, such as the torsion subgroup of any abelian group, has rank $0$.

Equivalently, the rank of $A$ is the dimension of the $\Q$-vector space $A \otimes_\Z \Q$. Rank is additive: if $0 \to A \to B \to C \to 0$ is exact, then $\rank B = \rank A + \rank C$. The free abelian group $\Z^n$ has rank $n$, and $\Q$ has rank $1$.