Almost self-injective, continuous, quasi-continuous (also known
as π-injective), and CS modules are generalizations of
injective modules. The main aim of this dissertation is to study
almost self-injective, continuous, quasi-continuous, and CS group
rings. CS group algebras were initiated by Jain et. al. They
showed that K[D∞] is a CS group algebra if and only if
char(K) ≠ 2. Behn extended this result and showed that if K[G]
is a prime group algebra with G polycyclic-by-finite, then K[G]
is a CS-ring if and only if G is torsion-free or G ≅ D∞ and char(K) ≠ 2. As a consequence, such a group
algebra K[G] is hereditary excepting possibly when K[G] is a
domain. We show that if K[G] is a semiprime group algebra of
polycyclic-by-finite group G and if K[G] has no direct summands
that are domains, then K[G] is a CS-ring if and only if K[G] is
hereditary if and only if G/Δ+(G)≅ D∞ and
char(K) ≠ 2. Furthermore, precise structure of a semiprime CS
group algebra K[G] of polycyclic-by-finite group G, when K is
algebraically closed, is also provided. Among others, it is shown
that (i) every almost self-injective group algebra with no
nontrivial idempotents is self-injective, (ii) if G is a torsion
group and the group algebra K[G] is quasi-continuous then G is
a locally finite group, and (iii) for any group G, if K[G] is
continuous then G is locally finite. As a consequence, it follows
that a CS group algebra K[G] is continuous if and only if K[G]
is principally self-injective if and only if G is locally
finite.
The properties of endomorphism rings of almost self-injective
indecomposable modules have been investigated. It is shown that the
endomorphism ring of a uniserial almost self-injective right module
is left uniserial. For a domain D, it is proved that D is right
almost self-injective if and only if D is a two sided valuation
domain.