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:

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

\(=\)

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

\(\ds \)

\(=\)

\(\ds \family {m_i}_{i \mathop \in I}\)

$\blacksquare$