  In this note we study associative dialgebras proving that the most
interesting such structures arise precisely when the algebra is not semiprime.
In fact the presence of some "perfection" property (simpleness, primitiveness,
primeness or semiprimeness) imply that the dialgebra comes from an associative
algebra with both products identified. We also describe the class of zero-cubed
algebras and apply its study to that of dialgebras. Finally we describe
two-dimensional associative dialgebras.
