  It is well known that a ring $R$ is right Kasch if each simple right
$R$-module embeds in a projective right $R$-module. In this paper we study the
dual notion and call a ring $R$ right dual Kasch if each simple right
$R$-module is a homomorphic image of an injective right $R$-module. We prove
that $R$ is right dual Kasch if and only if every finitely generated projective
right $R$-module is coclosed in its injective hull. Typical examples of dual
Kasch rings are self-injective rings, V-rings and commutative perfect rings.
Skew group rings of dual Kasch rings by finite groups are dual Kasch if the
order of the group is invertible. Many examples are given to separate the
notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings
are dual Kasch, and a commutative ring with finite Goldie dimension is dual
Kasch if and only if it is a classical ring (i.e. every element is a zero
divisor or invertible). We obtain that, for a field $k$, a finite dimensional
$k$-algebra is right dual Kasch if and only if it is left Kasch. We also
discuss the rings over which every simple right module is a homomorphic image
of its injective hull, and these rings are termed strongly dual Kasch.
