A new generalizations of retractable modules relative to a submodule are introduced where a module ℳ is called retractable module relative to a submodule N of M if for all non-zero submodule K of ℳ such that contains N, there exists a non-zero homomorphism f∈Hom(ℳ, K). Some basic properties are studied and many relationships between these classes and other related concepts are presented and studied.
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
In this work, the new notion is defined namely essentially compressible relative to a submodule, as a new generalization of the compressible module relative to a submodule where a module is called compressible module relative to a submodule N of M if for all non-zero submodule of such that contains N, there exists a monomorphism f Hom( ). We study some basic properties of this class and many relationships between these classes and other related concepts are presented and studied.
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