The ω-semimodules Over The Semidomains

In this thesis, we define the w-semimodules over the semidomains and introduce the concept of exact sequence in the category of R-semimodules. The exact sequence can be used to characterize the ω-semimodules. Moreover, the w-envelope of torsionfree semimodules is discussed in this thesis. All these explain that the definition of ω-semimodules forms a ω-operation over the semi-domains. Similar to the practice of domains and cancellable semigroups, SM semimodules, SM semidomains and Krull semidomains are defined. The torsion-free ω-semimodule M is a SM semimodule if and only if each subsemimodule of M is of finite type. The semidomain R is a SM semidomain if and only if each ideal of R is of finite type. At last, through introducing the concept of ω-invertible, we show that the semidomain R is a Krull semidomain if and only if each nonzero ideal is ω-invertible.