\section{Scalar Product with Multiple of Unity}
Tags: Unitary Modules

\begin{theorem}
Let $\struct {G, +_G}$ be an abelian group whose identity is $e$.
Let $\struct {R, +_R, \times_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $\struct {G, +_G, \circ}_R$ be an unitary $R$-module.
Let $x \in G, n \in \Z$.
Then:
:$\paren {n \cdot 1_R} \circ x = n \cdot x$
that is:
:$\paren {\map {\paren {+_R}^n} {1_R} } \circ x = \map {\paren {+_G}^n} x$
\end{theorem}

\begin{proof}
Follows directly from Scalar Product with Product.
{{qed}}
\end{proof}

