"Echelons and Janet bases" The classical division theorems for polynomials or power series provide an explicit description of direct complements of ideals in the ambient ring using leading or initial ideals. These complements are only vector spaces (generated by monomials), and thus leave the frame of ideals. Echelons are generalizations of ideals which admit direct complements which are again echelons, and which allow "free" generator systems (Janet bases). This approach is based on work of Riquier, Janet and Hironaka. Joint work with Mariemi Alonso and Francisco Castro.