The Characterization of Prime Submodules and Almost Prime Submodules of Finitely Generated Module over Principal Ideal Domain - GEDE ADHITYA WISNU WARDHANA (30112003), I
The abstraction of prime number in more complex mathematics structure has been begun by Dedekind in 1871. He introduced prime ideal as an abtraction of prime number in Ring Theory. In 1978 John Dauns gave generalization of prime number in Module Theory, named prime submodules. Hani A. Khashan introduced almost prime submodules a generalization of prime submodules in 2012. There are other generalization of prime submodules, such as weakly prime submodules that introduced by M.A. Hadi, or quasi prime submodules that introduced by Muntana. Khashan gave characterization of almost prime submodule in multiplicative module. Since direct sum of multiplicative module is not always a multiplicative module, we need to study characterization almost prime submodules in different situation. In this research we will investigate the characterization of prime submodules and almost prime submodules in finitely generated module over principal ideal domain. There are two methods that will be used: First we look at the pattern of almost prime submodules, and generalize it. Second we look at the properties of prime submodules and generalize it to almost prime submodule. The strategy of this research consists of three steps: First we will investigate the characterization of almost prime submodule in Z-module Zn. Next, we will investigate in free module over principal ideal domain and in torsion module over principal ideal domain. And last, we will investigate the characterization in finitely generated module over principal ideal domain.