Equational characterizations of cyclic purity and single compactness.

Abstract

The concepts cyclic purity and total purity for modules are defined by Simmons over commutative domains. In this work equational approaches are used to generalise these concepts and related results to modules over arbitrary rings. It is shown here that single splitness and single compactness as defined by Azumaya, coincide with these generalised versions of cyclic purity and total purity. Among other results, rings all of whose modules are totally pure are characterized, and coflatness is used to characterize rings over which every totally pure module is injective.