Definition:Closure (Abstract Algebra)/Scalar Product

Definition

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 $S$.

Also see

Some sources use stable for closed.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases