The quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring

Let G be a group. Let R be a G-graded commutative ring and let M be a graded R-module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever r is an element of h(R) and m is an element of h(M) with rm is an element of Q, then either r is an element of Gr((Q :(R) M)) or m is an element of Gr(M)(Q). The graded quasi-primary spectrum qp.Spec(g)(M) is defined to be the set of all graded quasi-primary submodules of M. In this paper, we introduce and study a topology on qp.Spec(g)(M), called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which (qp.Spec(g)(M), q.tau(g)) is a Noetherian, spectral space.