  The article focuses on four aspects related to the descent algebras of type
$A$. They are modular idempotents, higher Lie powers, higher Lie modules and
the right ideals of the symmetric group algebras generated by the Solomon's
descent elements. More precisely, we give a construction for the modular
idempotents, describe the dimension and character for higher Lie powers and
study the structures of the higher Lie modules and the right ideals both in the
ordinary and modular cases.
