  Let E be an elliptic curve having Complex Multiplication by the full ring O_K
of integers of K=Q(\sqrt{-D}), let H=K(j(E)) be the Hilbert class field of K.
Then the Mordell-Weil group E(H) is an O_K-module, and its structure denpends
on its Steinitz class St(E), which is studied here. In partucular, when D is a
prime number, it is proved that St(E)=1 if D\equiv 3 (mod 4); and St(E)=[P]^t
if D\equiv 1 (mod 4), where [P] is the ideal class of K represented by prime
factor P of 2 in K, t is a fixed integer. General structures are also discussed
for St(E) and for modules over Dedekind domain. These results develop the
results by D. Dummit and W. Miller for D=10 and some elliptic curves to more
general D and general elliptic curves.
