Let $R$ be a ring with identity, let $M$ be a left $R$-module, and let $(e_i)_{i \in I}$ be an indexed family of elements of $M$. Then $(e_i)_{i \in I}$ is a basis of $M$ if and only if the following two conditions hold:
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1. The family spans $M$: for every $m \in M$, there exist a finite subset $F \subset I$ and coefficients $(r_i)_{i \in F}$ in $R$ such that
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\begin{align*}
m = \sum_{i \in F} r_i e_i.
\end{align*}
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2. The family is $R$-linearly independent: for every finite subset $F \subset I$ and every family of coefficients $(r_i)_{i \in F}$ in $R$, if