COFINITELY 𝜹�−𝑯�− SUPPLEMENTED MODULES

Abstract

Consider a module 𝑃� over a ring 𝑆�. We describe 𝑃� as cofinitely 𝛿� − 𝐻� − 𝑠�upplemented, in case there is a direct summand 𝐾� of 𝑃� with the property that the equality 𝑃� = 𝐴� + 𝑋� holds if and only if 𝑃� = 𝐾� +𝑋� for any submodule 𝑋� of 𝑃� with singular 𝑃�/𝑋� and for each cofinite submodule 𝐴� of 𝑃�. In this work, we demonstrate that 𝑃� satisfies cofinitely 𝛿� − 𝐻� − 𝑠�upplemented condition if and only if 𝑃� has a direct summand 𝐾� with the properties (𝐴� + 𝐾�)/𝐴� ≪𝛿� 𝑃�/𝐴� and (𝐴� +𝐾�)/𝐾� ≪𝛿� 𝑃�/𝐾� for each cofinite submodule 𝐴� of 𝑃�. 𝛿� −semiperfect rings are characterized by means of cofinitely 𝛿� − 𝐻� −supplemented modules, with the characterization expressed through a set of equivalent statements.