Copure-injective modules
This article closely follows previous papers of the author [e.g., Acta Math. Hung. 44, 3-12 (1984; Zbl 0543.16016), Proc. Edinb. Math. Soc., II. Ser. 29, 289-298 (1986; Zbl 0582.16010) and Arch. Math. 48, 53-57 (1987; Zbl 0593.16018)]. An R-module A over a ring R with identity is said to be (i) cofinitely generated if its injective hull is the direct sum of the injective hulls of finitely many simple R-modules, (ii) cofree if it is isomorphic to a direct product of injective hulls of some family of simple R-modules, and (iii) cofinitely related if there is a short exact sequence \(0\to A\to B\to C\to 0\) where B is cofinitely generated, cofree and C is cofinitely generated. A short exact sequence of R-modules is copure if every cofinitely related R-module is injective relative to this sequence. Finally, an R-module A is copure-injective if it is injective relative to every copure short exact sequence. The main result of the paper states that, over any ring R, an R-module A is copure-injective if and only if it is the direct summand of a direct product of cofinitely related modules. These modules are then studied over classical rings, as defined by \textit{P. Vámos} [J. Algebra 34, 114-129 (1975; Zbl 0331.13006)], and Dedekind domains.
scientific article; zbMATH DE number 4093678