The study of neat homomorphisms
found in this paper originated with a generalization of neat subgroups and a property
of torsion free coverings studied by Enochs. Several useful characterizations and
properties of neat homomorphisms are shown leading to the characterization of
various rings. A ring is hereditary if and only if each component of the natural
decomposition of a neat homomorphism is neat. Furthermore, a ring is NQetherian if
and only if the direct sum (and direct limit) of a family of neat homomorphisms is
neat. For the singular torsion theory, every non-zert) torsion module contains a
simple submodule if and only if the product of a family of neat homomorphisms
is neat. If R has zero singular ideal and zero left socle, then the singular
theory coincides with the simple theory if and only if the above condition is
true.
