A. This is also known as ring extension A | B. Such a ring extension is called Frobenius if There is a linear mapping E: A → B satisfying the bimodule condition E(bac) = bE(a)c for all b,c ∈ B and a ∈ A. There are elements in A denoted and such that for all a ∈ A we have: The map E is sometimes referred to as a Frobenius homomorphism and the elements as dual bases. (As an exercise it is possible to give an equivalent definition of Frobenius extension as a Frobenius algebra-coalgebra object in the category of B-B-bimodules, where the equations just given become the counit equations for the counit E.) For example, a Frobenius algebra A over a commutative ring K, with associative nondegenerate bilinear form (-,-) and projective K-bases is a Frobenius extension A | K with E(a) = (a,1). Other examples of Frobenius extensions are pairs of group algebras associated to a subgroup of finite index, Hopf subalgebras of a semisimple Hopf algebra, Galois extensions and certain von Neumann algebra subfactors of finite index. Another source of examples of Frobenius extensions (and twisted versions) are certain subalgebra pairs of Frobenius algebras, where the subalgebra is stabilized by the symmetrizing automorphism of the overalgebra. The details of the group ring example are the following application of elementary notions in group theory. Let G be a group and H a subgroup of finite index n in G; let g1, ..., gn. be left coset representatives, so that G is a disjoint union of the cosets g1H, ..., gnH. Over any commutative base ring k define the group algebras A = k[G] and B = k[H], so B is a subalgebra of A. Define a Frobenius homomorphism E: A → B by letting E(h) = h for all h in H, and E(g) = 0 for g not in H : extend this linearly from the basis group elements to all of A, so one obtains the B-B-bimodule projection (The orthonormality condition follows.) The dual base is given by , since The other dual base equation may be derived from the observation that G is also a disjoint union of the right cosets . Also Hopf-Galois extensions are Frobenius extensions by a theorem of Kreimer and Takeuchi from 1989. A simple example of this is a finite group G acting by automorphisms on an algebra A with