\begin{definition}[Definition:Free Module on Set]
Let $R$ be a ring.
{{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, that is, complete with operators, $\struct {R, +, \circ}$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}}
Let $I$ be an indexing set.
The '''free $R$-module on $I$''' is the direct sum of $R$ as a module over itself:
:$\ds R^{\paren I} := \bigoplus_{i \mathop \in I} R$
of the family $I \to \set R$ to the singleton $\set R$.
\end{definition}