Let R be a G-graded ring. In this article, we introduce two new concepts on graded rings, namely, weakly rings and invertible discuss the relations between these several properties of rings. Also, study concept crossed products some defined product give relationship Moreover, in generalization for essential submodules, semi-uniform modules which is uniform modules.

In this paper, we complete the classification of the Z-graded modules of the intermediate series over the q-analog Virasoro-like algebra L. We first construct four classes of irreducible Z-graded L-modules of the intermediate series. Then we prove that any Z-graded L-modules of the intermediate series must be the direct sum of some trivial L-modules or one of the modules constructed by us.