Nearly Frobenius theory and semisimplicity of bimodules

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