  We define two classes of representations of quivers over arbitrary fields,
called monomorphic representations and epimorphic representations. We show that
every representation has a unique maximal nilpotent subrepresentation and the
associated quotient is always monomorphic, and every representation has a
unique maximal epimorphic subrepresentation and the associated quotient is
always nilpotent. The uniquenesses of such subrepresenations imply two
identities in the Ringel-Hall algebra. By applying Reineke's integration map,
we obtain two identities in the corresponding quantum torus.
