Frobenius Algebra - Properties

Properties

• The direct product and tensor product of Frobenius algebras are Frobenius algebras.
• A finite-dimensional commutative local algebra over a field is Frobenius if and only if the right regular module is injective, if and only if the algebra has a unique minimal ideal.
• Commutative, local Frobenius algebras are precisely the zero-dimensional local Gorenstein rings containing their residue field and finite dimensional over it.
• The right regular representation of a Frobenius algebra is always injective.
• For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Hom_k(A,k) is isomorphic to the right regular representation of A.
• For an infinite field k, a finite dimensional, unitial, associative k-algebra is a Frobenius algebra if it has only finitely many minimal right ideals.
• If F is a finite dimensional extension field of k, then a finite dimensional F-algebra is naturally a finite dimensional k-algebra via restriction of scalars, and is a Frobenius F-algebra if and only if it is a Frobenius k-algebra. In other words, the Frobenius property does not depend on the field, as long as the algebra remains a finite dimensional algebra.
• Similarly, if F is a finite dimensional extension field of k, then every k-algebra A gives rise naturally to a F algebra, F ⊗_k A, and A is a Frobenius k-algebra if and only if F ⊗_k A is a Frobenius F-algebra.
• Amongst those finite-dimensional, unital, associative algebras whose right regular representation is injective, the Frobenius algebras A are precisely those whose simple modules M have the same dimension as their A-duals, Hom_A(M,A). Amongst these algebras, the A-duals of simple modules are always simple.