Let $R$ be a unital ring with multiplicative identity $1_R$, let $M$ be a left unital $R$-module with zero element $0_M$, and let $S \subset M$. Let $\langle S \rangle_R$ denote the smallest submodule of $M$ containing $S$. Then
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\begin{align*}
\langle S \rangle_R = \left\{\sum_{i=1}^{n} r_i s_i : n \in \mathbb{Z}_{\ge 0},\ r_i \in R,\ s_i \in S \text{ for every } i \in \{1,\dots,n\}\right\}.
\end{align*}
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When $n = 0$, the sum is the empty sum and is equal to $0_M$. In particular,