  For the module category of an Artin algebra, we generalize the notion of
torsion pairs to ideal torsion pairs. Instead of full subcategories of modules,
ideals of morphisms of the ambient category are considered. We characterize the
functorially finite ideal torsion pairs, which are those fulfilling some nice
approximation conditions, first through corresponding functors and then through
the notion of ideals determined by objects introduced in this work. As an
application of this theory, we generalize preprojective modules, introduce a
new homological dimension, the torsion dimension, and establish its connection
with the Krull-Gabriel dimension. In particular, it is shown that both
dimensions coincide for hereditary Artin algebras.
