ADDED: The general structure of a group algebra (or other finite dimensional algebra) is studied in the traditional way using idempotents in the classical 1962 book by Curtis and Reiner Representations of Finite Groups and Associative Algebras. The idempotents generate left ideal summands (principal indecomposable modules) and survive in the semisimple quotient when the radical is factored out; sometimes these can be described explicitly, as in the case of symmetric groups. In your example of a family of groups of Lie type, split over the prime field, the structure of the group algebra over that field extends naturally to an algebraic closure where comparison with algebraic groups is possible.
In the case of a group algebra, the key information tends to involve the representation theory of the group over the given field. It may or may not be helpful to look for generators of the radical (as a two-sided ideal), but I don't know of any substantial results for this family of groups. The dimensions and module structure are transparent, however.