# Finite Direct Product of Unitary Modules is Unitary Module

## Theorem

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be unitary $R$-modules.

Let:

$\ds G = \prod_{k \mathop = 1}^n G_k$

be their direct product.

Then $G$ is a unitary module.

## Proof 1

This is a special case of Direct Product of Unitary Modules is Unitary Module.

## Proof 2

From Finite Direct Product of Modules is Module we have that $G$ is a module.

It remains to be shown that:

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

Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.

Then:

 $\ds 1_R \circ x$ $=$ $\ds 1_R \circ \tuple {x_1, x_2, \ldots, x_n}$ $\ds$ $=$ $\ds \tuple {1_R \circ x_1, 1_R \circ x_2, \ldots, 1_R \circ x_n}$ $\ds$ $=$ $\ds \tuple {x_1, x_2, \ldots, x_n}$ $\ds$ $=$ $\ds x$

Hence the result.

$\blacksquare$