Finite Direct Product of Modules is Module

Theorem
Let $$G_1, G_2, \ldots, G_n$$ be $R$-modules.

Let: $$G = \prod_{k=1}^n G_k$$

Then $$\left({G, +: \circ}\right)_R$$ is an $R$-module where:
 * $$+$$ is the operation induced on $$G$$ by the operations on $$G_1, G_2, \ldots, G_n$$;
 * $$\circ$$ is defined as $$\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}\right)$$

If each $$G_k$$ is a unitary module, then so is $$G$$.