Goldie rings graded by a group with periodic quotient group modulo the center

In this paper, we study gr-prime and gr-semiprime Goldie rings graded by a group with periodic quotient group modulo the center. We enhance the theorem of Goodearl and Stafford (2000) about gr-prime rings graded by Abelian groups; we extend the Abelian group class to the class of groups with periodic quotient group modulo the center. We also decompose the orthogonal graded completion $O^{\mathrm{gr}}(R)$ of a gr-semiprime Goldie ring $R$ (graded by a group satisfying the same condition) into a direct sum of gr-prime Goldie rings $R_1,\dots, R_n$ and prove that the maximal graded quotient ring $Q^{\mathrm{gr}}(R)$ equals the direct sum of classical graded quotients rings of $R_1,\dots, R_n$.