Modules that have a supplement in every cofinite extension. (Q2897175)

\textit{H. Zöschinger} [in his paper Math. Scand. 35(1974), 267-287 (1975; Zbl 0299.13006)] studied modules \(M\) with the following properties: (E): \(M\) has supplement in every extension; (EE): \(M\) has ample supplements in every extension.NEWLINENEWLINE By introducing the notion of cofinite extension the authors, in this paper, study modules \(M\) with the following properties: (CE): \(M\) has supplement in every cofinite extension; (CEE): \(M\) has ample supplements in every cofinite extension.NEWLINENEWLINE They prove that a module \(M\) has property (CEE) if and only if every submodule of \(M\) has property (CE). They prove that the property (CE) is preserved under direct summands and extensions. They prove that a ring \(R\) is semiperfect if and only if every \(R\)-module has property (CE) ((CEE)). The authors conclude the paper with an example to show that the property (CE) does not imply the property (E).