Goldie Rings Graded by a Group with Periodic Quotient Group Modulo the Center

Abstract

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^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 R₁, . . . , R_n and prove that the maximal graded quotient ring Q^gr(R) equals the direct sum of classical graded quotients rings of R₁, . . . , R_n.