\begin{definition}[Definition:Direct Sum of Modules]
Let $A$ be a commutative ring with unity.
{{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, $\struct {A, +, \circ}$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}}
Let $\family {M_i}_{i \mathop \in I}$ be a family of $A$-modules indexed by $I$.
{{explain|Worth specifically stating exactly what $I$ is.}}
Let $M = \ds \prod_{i \mathop \in I} M_i$ be their direct product.
The '''direct sum''' $\ds \bigoplus_{i \mathop \in I} M_i$ is the submodule of $M$ the consisting of the elements of finite support.
\end{definition}