  This article is divided into two parts. In the first part we work over a
field $\mathbb{k}$ 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 $k$. We
introduce the concept of nearly Frobenius algebras in this context and
construct solutions of the Yang-Baxter equation starting from elements in the
Frobenius space. Also, we give a list of equivalent characterizations of nearly
Frobenius algebras.
