\begin{definition}[Definition:Linearly Independent/Sequence]
Let $G$ be an abelian group whose identity is $e$.
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.
Let $\sequence {a_n}$ be a sequence of elements of $G$ such that:
:$\ds \forall \sequence {\lambda_n} \subseteq R: \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0_R$
That is, the only way to make $e$ with a linear combination of $\sequence {a_n}$ is by making all the terms of $\sequence {\lambda_n}$ equal to $0_R$.
Such a sequence is '''linearly independent'''.
\end{definition}