Let $\{ X_i : i \in I \}$ be a collection of modules in some category of modules. Then the direct product $\prod_{i \in I} X_i$ of that collection is the module whose underlying set is the Cartesian product of the $X_i$ with componentwise addition and scalar multiplication. For example, in a category of left modules: $$(x_i) + (y_i) = (x_i + y_i),$$ $$r (x_i) = (r x_i).$$

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

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

The direct product is often referred to as the complete direct sum, or the strong direct sum, or simply the product.

Compare this to the direct sum of modules.