Minimal Cogenerators Over Osofsky and Camillo Rings

The direct sum C of the injective hulls E(V _i ) of the set {V _i } _{i ∈ I} of non-isomorphic simple right R-modules is a minimal right co-generator for R. While the injective hull E(C) is the unique (up to isomorphism) minimal injective right cogenerator, Osofsky [0] showed C is not necessarily unique even for commutative R, but that it is when R is either right Noetherian, semilocal, or C is quasiinjective. In this paper, we call a ring R a right Osofsky ring when C is the unique minimal right cogenerator, and show that rings studied by Camillo [Cl] with the property that Hom _R (E(V _i ), E(V _j )) = 0 for i ≠ j, are right Osofsky. We call these right Camillo rings, and show that commutative SISI rings of Vámos [V], and locally perfect commutative rings, in fact, any 0-dimensional ring, among others, are Camillo, hence Osofsky rings.