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A module M over a associative ring with unity R is called quasi-Dedekind if every nonzero submodule of M is quasi-invertible (Equivalently a module M is quasi-Dedekind if for any nonzero homomorphism f from M to itself, f is monomorphism). In this book we study essentially quasi-Dedekind modules as a generalization of quasi-Dedekind modules, where an R-module M is said to be essentially quasi-Dedekind if every essential submodule of M is quasi-invertible. We prove that this concept is equivalent to the concept K-nonsingular modules, where a module M is said to be K-nonsingular if for each nonzero homomorphism f from M to itself implies Kerf is not essential in M. We studied the relationships between this concept and several other types of modules such as: quasi-Dedekind, prime, essentially prime, scalar, nonsingular, monoform, Baer and compressible modules,... etc. Beside, we present the concept of small quasi-Dedekind module as another generalization of quasi-Dedekind module, where an R-module M is called small quasi-Dedekind if for each nonzero homomorphism f from M to itself, Kerf is small in M.
About the author
Tha'ar Younis Ghawi is teacher at the Al-Qadisiyah University in Iraq. He received his ph.D degree in Algebra/ Modules theory in (2015) from Al-Mustansiriyah University/ Iraq. Ghawi wrote many papers in the area of modules theory.
