Definition:Subalgebra

Definition
Let $R$ be a commutative ring.

Let $\left({A, *}\right)$ be an algebra over $R$.

A subalgebra of $A$ is a submodule $B \subseteq A$ such that:
 * $\forall x, y \in B: x * y \in B$

That is, such that $B$ is closed under $*$.

Also see

 * Definition:Unital Subalgebra