  Recently, in a series of papers "simple" versions of direct-injective and
direct-projective modules have been investigated. These modules are termed as
"simple-direct-injective" and "simple-direct-projective", respectively. In this
paper, we give a complete characterization of the aforementioned modules over
the ring of integers and over semilocal rings. The ring is semilocal if and
only if every right module with zero Jacobson radical is
simple-direct-projective. The rings whose simple-direct-injective right modules
are simple-direct-projective are fully characterized. These are exactly the
left perfect right $H$-rings. The rings whose simple-direct-projective right
modules are simple-direct-injective are right max-rings. For a commutative
Noetherian ring, we prove that simple-direct-projective modules are
simple-direct-injective if and only if simple-direct-injective modules are
simple-direct-projective if and only if the ring is Artinian. Various closure
properties and some classes of modules that are simple-direct-injective (resp.
projective) are given.
