Finite_Direct_Product_of_Modules_is_Module_v2

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be $R$-modules. Let $\ds G = \prod_{k \mathop = 1}^n G_k$ be their direct product. Then $G$ is a module.