In this paper we develope a Morsification Theory for holomorphic functions
defining a singularity of finite codimension with respect to an ideal, which
recovers most previously known Morsification results for non-isolated
singulatities and generalize them to a much wider context. We also show that
deforming functions of finite codimension with respect to an ideal within the
same ideal respects the Milnor fibration. Furthermore we present some
applications of the theory: we introduce new numerical invariants for
non-isolated singularities, which explain various aspects of the deformation of
functions within an ideal; we define generalizations of the bifurcation variety
in the versal unfolding of isolated singularities; applications of the theory
to the topological study of the Milnor fibration of non-isolated singularities
are presented. Using intersection theory in a generalized jet-space we show how
to interprete the newly defined invariants as certain intersection
multiplicities; finally, we characterize which invariants can be interpreted as
intersection multiplicities in the above mentioned generalized jet space.Comment: 56 pages, some typos correcte