Let $\{ X_i : i \in I \}$ be a collection of modules in some category of modules. Then the direct sum $\coprod_{i \in I} X_i$ of that collection is the submodule of the direct product of the $X_i$ consisting of all elements $(x_i)$ such that all but a finite number of the $x_i$ are zero.

For each $j \in I$ we have a projection $p_j : \coprod_{i \in I} X_i \to X_j$ defined by $(x_i) \mapsto x_j$ , and an injection $\lambda_j : X_j \to \coprod_{i \in I} X_i$ where an element $x_j$ of $X_j$ maps to the element of $\coprod_{i \in I} X_i$ whose $j$ th term is $x_j$ and every other term is zero.

The direct sum $\coprod_{i \in I} X_i$ satisfies a certain universal property. Namely, if $Y$ is a module and there exist homomorphisms $f_i : Y \to X_i$ for all $i \in I$ , then there exists a unique homomorphism $\phi : \coprod_{i \in I} X_i \to Y$ satisfying $p_i \phi = f_i$ for all $i \in I$ .

The direct sum is often referred to as the weak direct sum or simply the sum.

Compare this to the direct product of modules.

Often an internal direct sum is written as $\bigoplus_{i \in I} X_i$ .