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 compressiblemodules was introduced in 1981 by Zelmanowitz, where module M is called compressible if it can be embedded inany nonzero submodule A of M . In other words, M is a compressible module if for each nonzero submodule A ofM, f 2 Hom(M;A) exists, such that f is monomorphism. Retractable modules were introduced in 1979 Khuri, wheremodule 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 modulesrelative to a submodule are introduced, where module M is called compressible module relative to a submoduleN of M . If for all nonzero submodule K of M contains N , then a monomorphism f 2 Hom(M, K) exists. Somebasic properties are studied and many relationships between these classes and other related concepts are presentedand studied. We also introduce another generalization of retractable module, which is called small kernel retractablemodule