Direct Product of Unitary Modules is Unitary Module

Theorem

Let $R$ be a ring with unity whose unity is $1_R$.

Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of unitary $R$-modules.

Let $\struct {M, +, \circ}$ be their direct product.

Then $\struct {M, +, \circ}$ is a unitary $R$-module.

It remains to verify that:

$\forall x \in M: 1_R \circ x = x$

By the definition of direct product, $\circ$ is an $R$-action induced on $M$ by $\family {\circ_i}_{i \mathop \in I}$:

$\forall r \in R: r \circ \family {m_i}_{i \mathop \in I} = \family {r \circ_i m_i}_{i \mathop \in I}$.

We have:

$\ds \exists 1_R \in R, \forall \family {m_i}_{i \mathop \in I} \in M: \, $

$\ds 1_R \circ \family {m_i}_{i \mathop \in I}$

$=$

$\ds \family {1_R \circ_i m_i}_{i \mathop \in I}$

Definition of $R$-action

$\ds $

$=$

$\ds \family {m_i}_{i \mathop \in I}$

$\blacksquare$