International Journal of Algebra, Vol. 1, 2007, no. 12, 601 - 613 Cofinitely δ -Supplemented and Cofinitely δ -Semiperfect Modules Khaled Al-Takhman Mathematics Department, Birzeit University Birzeit, P.O. Box 14, Palestine takhman@birzeit.edu Abstract In this work, we prove that an R-module M is cofinitely δ-supple- mented (i.e. each cofinite submodule of M has a δ-supplement) if and only if every maximal submodule of M has a δ-supplement. An R- module M is called cofinitely δ-semiperfect if each finitely generated factor module of M has a projective δ-cover, we prove that this is equiv- alent to the existence of a δ-supplement, which is a direct summand of M , for each cofinite submodule of M . Cofinitely δ-lifting modules are introduced and characterized. We also give new characterizations of δ- semiperfect rings in terms of these concepts. Some examples are given at the end of this article. Mathematics Subject Classification: 16L30, 16E50 Keywords: cofinitely δ -supplemented, cofinitely δ -semiperfect modules, projective δ -cover, δ -semiperfect rings 1 Introduction And Preliminary Notes Throughout this paper R is an associative ring with unity and all modules are unitary left R-modules. A submodule K of a module M is denoted by K ⊆ M . Let M be a module, K ⊆ M is called small in M (denoted K ≪ M ) if for every N ⊆ M , the equality N + K = M implies N = M . A submodule U ⊆ M is called a supplement of K ⊆ M , if M = K + U and K ∩ U ≪ U . Zhou [12] introduced the concept of ”δ -small submodule” as a general- ization of small submodules. Let K ⊆ M , K is called δ -small in M , de- noted by K ≪ δ M , if whenever M = N + K and M/N is singular, we have M = N . The sum of all δ -small submodules of a module M is denoted by