Nearly Frobenius dimension of Frobenius algebras

This article is divided into two parts. In the first part we work over a field k and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. In particular, we prove that the Frobenius dimension coincides with the dimension of the algebra. In the second part we work with a commutative ring k and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. We introduce the concept of nearly Frobenius algebras in this context and construct solutions of the quantum Yang-Baxter equation starting from elements in the Frobenius space. Also, we give an alternative characterization of a nearly Frobenius algebra.