Definition:Module Direct Product
Definition
Finite Case
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 $M_n$.
General Case
Let $R$ be a ring.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of $R$-modules.
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Let:
- $\ds M = \prod_{i \mathop \in I} M_i$
be the cartesian product of these modules.
The operation $+$ induced on $M$ by $\family {+_i}_{i \mathop \in I}$ is the operation defined by:
- $\family {a_i}_{i \mathop \in I} + \family {b_i}_{i \mathop \in I} = \family {a_i +_i b_i}_{i \mathop \in I}$
That is, the additive group of the module $M$ is the direct product of the groups $\family {\struct {M_i, +_i} }_{i \mathop \in I}$.
The $R$-action $\circ$ induced on $M$ by $\family {\circ_i}_{i \mathop \in I}$ is the operation defined by:
- $r \circ \family {m_i}_{i \mathop \in I} = \family {r \circ_i m_i}_{i \mathop \in I}$
In Direct Product of Modules is Module, it is shown that $\struct {M, +, \circ}$ is an $R$-module.
The module $\struct {M, +, \circ}$ is called the (external) direct product of $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$.