  In this paper we define the modified Ringel-Hall algebra $\cm\ch(\ca)$ of a
hereditary abelian category $\ca$ from the category $C^b(\mathcal{A})$ of
bounded $\mathbb{Z}$-graded complexes. Two main results have been obtained. One
is to give a new proof of Green's formula on Ringel-Hall numbers by using the
associative multiplication of the modified Ringel-Hall algebra. The other is to
show that in certain twisted cases the derived Hall algebra can be embedded in
the modified Ringel-Hall algebra. As a consequence of the second result, we get
that in certain twisted cases the modified Ringel-Hall algebra is isomorphic to
the tensor algebra of the derived Hall algebra and the torus of acyclic
complexes and so the modified Ringel-Hall algebra is invariant under derived
equivalences.
