The Characterization Of Generators And Cogenerators In The Category Of Semimodules

This passage studies the structure and properties of semimodule in semiring. Infrastructure of semimodule is commutative monoid semigroup, and its "factor" section is semiring. Therefore, the properties of the semimodule is essentially different with module, given semimodule homomorphism f :M→N, k f = 0 is a necessary condition for monomorphism but not sufficient condition, Im f = N is sufficient condition but not necessary conditions for epimorphism. So research of semimodule is much more difficult than module.This passage studies the generator and congenerator in semimodule category. And the exact sequence and proper exact sequence is introduced to describe the properties of generator and cogenerator . This passage is divided into four chapters:In chapter1,we give the introduction and basic notion and terminologies.In chapter2, the concept of generator in semimodule category is first given and we get the equivalence relation of generator , thus there is a class of dyadic relation that exists between generator and projective semimodules.In chapter3, the concept of congenerator in semimodule category is first given and we get the equivalence relation of congenerator .we get maintain the proper exact cooperation of Hom functor.In chapter4, the concept of annihilator in semimodule category is first given and we get the equivalence relation of annihilator, generator and congenerator. And the concept of the trace and reject is introduced, the its properties are discussed , there is a good description of the module and generator .