Prime Submodules Over Commutative Rings

In this paper, we study the prime submodules systematically by using thew-operation and utilizing the module theory method. In the first part of eachchapter, we give some basic characterizations of prime submodules or prime w-submodules. First, we discuss the Principal Ideal Theorem for modules throughbasic properties and conclusions of prime submodules. Some equivalent descrip-tions are given for the PIT to be hold for projective modules over SM domains.It is proved that the PIT holds for every projective R-module if and only ifthe PIT holds for R⁽²⁾, if and only if the local ring R_p is a valuation ring forevery height 1 prime ideal p of R, if and only if the local ring R_p is a discretevaluation ring for every height 1 prime ideal p of R. Meanwhile, we give someequivalent characterizations of GV-ideals over unique factorization domains. LetR be a unique factorization domain, then I = Rα₁ +…+ Rα_n∈GV(R) if andonly if N = R(a₁,…,a_n) is a prime submodule of F = R⁽ⁿ⁾ (n≥2) such thatrank(N) = 1, if and only if N = R(α₁,…,a_n) is a submodule of F = R⁽ⁿ⁾ (n 2)which is maximal of rank(N) = 1. Secondly, we define the concept of w-radicalof a submodule of a w-module, and discuss its basic properties. As applicationsof the obtained results, we study the w-radicals over Laskerian modules, multi-plicative modules and finitely generated free modules respectively. It is provedthat if M is a finite type multiplicative w-module and I is a w-ideal of R, thenw-radIM = I^1/2M. It is discussed the w-radical of the direct sum of submodules,and proved that if {M_i|i∈Γ} are a family of w-modules, Ni(i∈Γ) are the sub-modules of Mi, and M =⊕M_i, N =⊕N_i, then w-rad_M N =⊕w-rad_Mi N_i.Moreover, it is proved that if F = R⁽ⁿ⁾, andα∈F, thenα∈w-radF N if and only if [α,α₁,…,α_m]_t (?) w-radR [0,α₁,…,α_m]t for all1≤t≤min{m + 1,n}.