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HONAM MATHEMATICAL JOURNAL, vol.45, no.1, pp.146-159, 2023 (ESCI)
In this paper, we define circle plus delta-co-coatomically supplemented and co-coatomically 6-semiperfect modules as a strongly notion of circle plus-co-coatomically supplemented and co-coatomically semiperfect modules with the help of Zhou's radical. We say that a module A is circle plus delta-co-coatomically supplemented if each co-coatomic submodule of A has a 6-supplement in A which is a direct summand of A. And a module A is co-coatomically 6-semiperfect if each coatomic factor module of A has a projective 6-cover. Also we define co-coatomically amply 6-supplemented modules and we examined the basic properties of these modules. Further-more, we give a ring characterization for our modules. In particular, a ring R is 6-semiperfect if and only if each free R-module is co-coatomically 6-semiperfect.