On Almost Maximal Submodules

The concepts of maximal ideals, maximal submodules are well-known in ring (module) theory. Here we introduce and study the concepts of almost maximal ideals, and almost maximal submodules. We define a proper submodule N of an R-module M to be an almost maximal submodule if whenever w is an essential submodule of M, N subset W subseteq M, implies W = M. A proper ideal I of a ring R is said to be an almost maximal ideal if it is an almost maximal R-submodule of R. Various properties of almost maximal submodules (ideals) are considered.