Definition:Module Direct Product/Finite Case

Definition

Let $R$ be a ring.

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

Let:

$\ds M = \prod_{k \mathop = 1}^n M_k$

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 $\lambda \circ \tuple {x_1, x_2, \ldots, x_n} = \tuple {\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_n x_n}$

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