Let $k$ be a ring and suppose $M$ is a module over $k$. A set $X \subseteq M$ is a minimal generating set if it generates $M$ and no proper subset of $X$ generates $M$.
It is easy to see this means that no element of $X$ can be written as a finite $k$-linear combination of the other elements in $X$. However this does NOT correspond to "linear independence" as is the case for vector spaces. For example if you consider $\mathbb Z_3$ as a $\mathbb Z$-module then $\{ 1 \}$ is a minimal generating set but $ 6 \cdot 1 = 3 \cdot 1 = 0$ but $3 \not= 6$.
However don't these notions coincide when we look at modules over a field, i.e. vector spaces? Why does $ \sum \alpha_i x_i = 0 \implies $ every $\alpha_i = 0$ if $\{x_i \}$ is a basis for a space $V$, but it doesn't hold for arbitrary modules?