= Free module
{title2=$M\cong\bigoplus_{i\in I}R$}
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A free module is a <module> with a <basis of a module> $(e_i)_{i\in I}$: every element has exactly one expression $m=\sum_{i\in I}r_ie_i$ with $r_i\in R$ and only finitely many $r_i$ nonzero. Equivalently, the coefficient map $\bigoplus_{i\in I}R\to M$ is a <module isomorphism>. Existence means generation; uniqueness means <linear independence in a module>. For finite $I$ of size $n$, this says $M\cong R^n$, a <finite free module>. See https://neilstrickland.github.io/modules/notes/modules.pdf[Neil Strickland's Rings, Modules and Linear Algebra, section 11].
The <integer lattice> $\mathbb Z^2$ in the opening diagram is a <free module> over $\mathbb Z$ with <basis of a module> $e_1=(1,0)$, $e_2=(0,1)$. For example, $(3,2)=3e_1+2e_2$, and comparing the two coordinates makes the coefficients unique. A different <basis of a module> is $u=(1,0)$, $v=(1,1)$, because $(a,b)=(a-b)u+bv$; in particular $(3,2)=u+2v$.
By contrast, $w_1=(2,0)$ and $w_2=(0,1)$ are <linearly independent in a module>, but generate only the <submodule> $2\mathbb Z\times\mathbb Z$. They miss $(1,0)$ because $2a=1$ has no <integer> solution. They would form a <basis> over the <rational numbers>. Even generation need not let one extract a <basis of a module>: the elements $2,3$ generate $\mathbb Z$, since $1=3-2$, but neither singleton generates it, and $3\cdot2-2\cdot3=0$ is a nontrivial <linear relation> between them. These are concrete failures of the usual <vector space> argument that divides by any nonzero coefficient.
The <cyclic module> $\mathbb Z/6\mathbb Z$ has no <basis of a module> over $\mathbb Z$: every element has the nontrivial relation $6m=0$, so no nonempty subset is <linearly independent in a module>, while the empty subset cannot generate this nonzero <module>. Specifying the scalar <ring> is essential: over $R=\mathbb Z/6\mathbb Z$, the same additive object is a <free module> with the one-element <basis of a module> $\{\overline1\}$, since each coefficient is itself a unique class in $R$.
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