Definition:Subalgebra

Definition

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$.

Let $B_R \subseteq A_R$ such that:

$\forall x, y \in B_R: x \oplus y \in B_R$

That is, that $\oplus$ is closed in $B_R$.

Then $B_R$ is a subalgebra of $A_R$.