Associative Dialgebras from a structural viewpoint

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.