  A ring $R$ is called right (small) dual if every (small) right ideal of $R$
is a right annihilator. Left (small) dual rings can be defined similarly. And a
ring $R$ is called (small) dual if $R$ is left and right (small) dual. It is
proved that $R$ is a dual ring if and only if $R$ is a semilocal and small dual
ring. Several known results are generalized and properties of small dual rings
are explored. As applications, some characterizations of QF rings are obtained
through small dualities of rings.
