The *_w-Operation And *-UMT Domains

Star-operations have received a good deal of attention continuously in anumber of literature in recent decade years. In this paper, we study *_w-ideals ofpolynomial rings, P*_tMD and *-UMT domains mainly by utilizing *_w-operations.Firstly, we discuss the properties of *_w-operations in modules and prime submod-ules. Using ideal theory methods, we prove that M is a torsion-free module, Aand B are submodule of M, A_*w = B_*w if and only if for every m∈*_w-Max(R),A_m = B_m. Using module theory methods, we prove that every projective mod-ule is *_w-module. Furthermore, we prove if F is a ?at module, it is *_w-module.Secondly, we study *_w-ideals of polynomial rings and UTZ mainly by utilizing*_w-operations. We prove that Q is a maximal *_w-ideal of R[X], p = Q∩R≠0,then Q = p[X], and p is maximal *_w-ideal of R. We prove that p is UTZ in R[X],then p is *_w-reversible ideal if and only if p is maximal *_w-ideal if and only ifc(p)*_w = R if and only if c(p) is *_w-reversible ideal if and only if there exists g∈psuch that c(g)_*w = R. We introduce the notion of *_w-domain, and prove that I isan ideal in R, I_*w (?) (IT)_*W if and only if J∈* - GV (R), then JT∈* - GV (T)if and only if for every prime *_w-ideal P in T, P∩R is *_w-ideal in R if and only ifT is a *_w-domain of R. Moreover, we introduce the notion of P*MD and *-UMTdomains, and prove that R is a P*_tMD domain if and only if R is a P*MD domainif and only if R is a P*_wMD domain, and prove *_w = *_t for the *-UMT domains.We prove that R is a *-UMT domains, p is a prime *_w-ideal in R, T is a domainand T is algebraic extension of R, Q is T a primary ideal and Q∩R = p, then Qis *_w-ideal of T .