# Direct Product of Unitary Modules is Unitary Module

## Theorem

Let $R$ be a ring with unity.

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.

## Proof

From Direct Product of Modules is Module, $M$ is an $R$-module.

It remains to verify that:

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

We have:

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

$\blacksquare$