## Injectivity, continuity, and CS conditions on group rings

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##### Abstract

Almost self-injective, continuous, quasi-continuous (also known as 7r-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 [24]. They showed that K[D1] is a CS group algebra if and only if char(K) =6 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 '= D1 and char(K) =6 2 [5]. 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/A+(G) ~= D1 and char(K) =6 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 unis¬erial 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.