Definition:Module Direct Product/Finite Case

Definition
Let $R$ be a ring.

Let $\left({M_1, +_1, \circ_1}\right)_R, \left({M_2, +_2, \circ_2}\right)_R, \ldots, \left({M_n, +_n, \circ_n}\right)_R$ be $R$-modules.

Let:
 * $\displaystyle M = \prod_{k \mathop = 1}^n M_k$

be the cartesian product of $M_1$ to $M_n$.

Let:
 * $+$ be the operation induced on $M$ by the operations $+_1, +_2, \ldots, +_n$ on $M_1, M_2, \ldots, M_n$


 * $\circ$ be defined as $\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_n x_n}\right)$

In Finite Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module.

The module $\left({M, +, \circ}\right)_R$ is called the (external) direct product of $M_1$ to $M_n$.