Essentially Retractable Modules Relative To A Submodule And Some Generalizations

R is a ring with unity, and all modules are unitary right R-modules. The concept of compressible modules was introduced in 1981 by Zelmanowitz, where module M is called compressible if it can be embedded in any nonzero submodule A of M . In other words, M is a compressible module if for each nonzero submodule A of M, f 2 Hom(M;A) exists, such that f is monomorphism. Retractable modules were introduced in 1979 Khuri, where module M is retractable if Hom(M, A ) 6= 0 for every nonzero submodule A of M . We define a new notion, namely, essentially retractable module relative to a submodule. In addition, new generalizations of compressible modules relative to a submodule are introduced, where module M is called compressible module relative to a submodule N of M . If for all nonzero submodule K of M contains N , then a monomorphism f 2 Hom(M, K) exists. Some basic properties are studied and many relationships between these classes and other related concepts are presented and studied. We also introduce another generalization of retractable module, which is called small kernel retractable module