On weakly prime radical

If T is an S- module and Q is a submodule of T, which is proper, then Q is called prime if xn∈Q implies n∈Q or xT⊆Q for some x∈S, n∈T. Also, if xym∈Q implies xm∈Q or ym∈Q for some x,y∈S, m∈T, then Q is called weakly prime submodule. One can easily show that prime submodules are weakly prime. We get some properties of weakly prime radical which are always true for prime radical. (Q∶T) is always a prime ideal when Q is a prime submodule. We have shown that if Q is a weakly prime submodule, (Q∶m) is a prime ideal for every m∈T-Q. In this thesis, we give the definition of a weakly quasi-primary submodule which generalizes the concept of a weakly primary submodule. Also we show that every weakly quasi-primary submodule Q is weakly prime if and only if 〈E_T (Q)〉=Q. If S is a commutative ring with identity whose prime ideals are totally ordered, then it is shown that a weakly prime radical is a weakly prime submodule, and the weakly radical formula holds for S. Finally, we prove that divided domains satisfy weakly radical formula.