1997 | OriginalPaper | Chapter
Minimal Cogenerators Over Osofsky and Camillo Rings
Author : Carl Faith
Published in: Advances in Ring Theory
Publisher: Birkhäuser Boston
Included in: Professional Book Archive
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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.