nLab subalgebra

A subalgebra is a subobject in a category of algebras.

That is to say, a subalgebra of an associative algebra $A$ is a subset of the underlying set of $A$ that is closed under the operations of addition and multiplication, and contains all elements of the base ring.

For other types of algebras, such as algebras over operads, algebras for endofunctors, algebras over monads, algebras over algebraic theories, the notion of subalgebra is defined as a subobject in the corresponding category of algebras.