algebra. The A homomorphism from xA into A induced by x ↦ y cannot be extended to an A homomorphism from A into A, showing that the ring is not self-injective, thus not Frobenius. Any finite-dimensional Hopf algebra, by a 1969 theorem of Larson-Sweedler on Hopf modules and integrals. 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. Frobenius algebras are quasi-Frobenius rings, and in particular, they are left and right Artinian and left and right self-injective. For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Homk(A,k) is isomorphic to the right regular representation of A. For an infinite field k, a finite-dimensional, unital, 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, HomA(M,A). Amongst these algebras, the A-duals of simple modules are always simple. A finite-dimensional bi-Frobenius algebra or strict double Frobenius algebra is a k-vector-space A with two multiplication structures as unital Frobenius algebras (A, • , 1) and (A, , ): there must be multiplicative homomorphisms and of A into k with and non-degenerate, and a k-isomorphism S of A onto itself which is an anti-automorphism for both structures, such that This is the case precisely when A is a finite-dimensional Hopf algebra over k and S is its