subalgebra

A subalgebra of an algebra A over a field F is a subset B of A that is itself an F-algebra with the operations inherited from A. Equivalently, B is a subspace of A over F that is closed under the product of A. If A is unital, a commonly used convention is that B contains the unit 1_A; some authors do not require this.

Consequences and scope: The subalgebras of A form a lattice under inclusion; the intersection of any collection of subalgebras is again a subalgebra. The subalgebra generated by a subset S of A is the smallest subalgebra containing S, equal to the intersection of all subalgebras containing S. For a homomorphism φ: A → C, the image φ(B) of a subalgebra B is a subalgebra of C.

Examples: In the matrix algebra M_n(F), the set of diagonal matrices is a subalgebra; the set of upper triangular matrices is also a subalgebra. The collection of scalar matrices {λI} forms a subalgebra isomorphic to F. The polynomial algebra F[x] can be embedded into any F-algebra via a chosen element, and the subalgebra generated by that element is F[A] = {p(a) : p ∈ F[x]}.

Variants and remarks: In ring theory, the term subalgebra is sometimes used for subrings that share the same identity; in other contexts, a subalgebra need not contain the identity. Subalgebras are fundamental in studying structure, representations, and invariants of algebras.