\begin{definition}[Definition:Closure (Abstract Algebra)/Scalar Product]
Let $\struct {S, \circ}_R$ be an $R$-algebraic structure over a ring $R$.
Let $T \subseteq S$ such that $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$.
Then $T$ is '''closed for scalar product'''.
If $T$ is also closed for operations on $S$, then it is called a '''closed subset''' of $S$.
\end{definition}