  In the first part of this article we prove that one of the conditions
required in the original definition of nearly Frobenius algebra, the
coassociativity, is redundant. Also, we determine the Frobenius dimension of
the product and tensor product of two nearly Frobenius algebras from the
Frobenius dimension of each of them. We apply these results to semisimple
algebras.
  In the second part we introduce the notion of normalized nearly Frobenius
algebra. We prove a series of equivalences: the concept of normalized nearly
Frobenius algebra is equivalent to the concept of separable algebra, equivalent
to the fact that the algebra is projective as a bimodule on itself and,
finally, equivalent to the category of bimodules is semisimple. Also, we relate
these concepts with the property of semisimplicity of the category of modules
over the algebra.
