A variation of supplemented modules

F-w-Supplemented Modules

“…Recently, several authors have studied different generalizations of small submodules (see, for example, [1], [2], [7], [8]). In [2], F-small submodules are defined and studied as a generalization of small submodules.…”

“…The submodule 𝐻 ⊆ 𝐸 has ample Fsupplements if, for every 𝐿 ⊆ 𝐸 with 𝐻 + 𝐿 = 𝐸, there is an F-supplement 𝐺 of 𝐻 with 𝐺 ⊆ 𝐻. The generalization of 𝑅𝑎𝑑(𝐸) is defined as 𝑅𝑎𝑑 𝐹 (𝐸) that is, intersection of whole maximal submodules of 𝐸 that contains 𝐹 in [1]. In [5], 𝐹small submodules and F-supplement submodules are investigated, and some properties of 𝐹supplement submodules which are adapted from supplement submodules are given.…”

“…In [5], 𝐹small submodules and F-supplement submodules are investigated, and some properties of 𝐹supplement submodules which are adapted from supplement submodules are given. In [1], another variation of supplemented modules is defined. The module 𝐸 is said to be w-supplemented if each semisimple submodule of 𝐸 has a supplement in 𝐸.…”