A C *-algebra A is called an ideal C * -algebra (or equally a dual algebra) if it
is an ideal in its bidual A**. M.C.F. Berglund proved that subalgebras and
quotients of ideal C*-algebras are also ideal C*-algebras, that a commutative
C *-algebra A is an ideal C *-algebra if and only if it is isomorphicto C (Q) for
some discrete space ?. We investigate ideal J*-algebras and show that the
above results can be generalized to that of .I*-algebras. Furthermore, it is
proved that if A is an ideal ,J*-algebra, then sp(a* a) has no nonzero limit point
for each a in A and consequently A has semifinite rank and is a restricted
product of its simple ideals