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We present two criteria for a group $G$ to satisfy the following statements:any $G$-graded gr-prime (gr-semiprime) right gr-Goldie ring admits agr-semisimple graded right classical quotient ring. The criterion forgr-semiprime rings is that the group $G$ is periodic. Actually, the sufficiencyof periodicity was proved by the author in 2011 and the necessity of it followsfrom the well-known counterexample (1979). The main result of the paperconcerns the gr-prime case. In this case, Goodearl and Stafford proved thegraded version of Goldie's Theorem for rings graded by an Abelian group (2000).Developing their idea we extend the class of groups to the following: $(\forallg,h\in G)(\exists n\in \mathbb{N})(gh^n=h^ng)$. Moreover, for any group $G$outside this class, we construct a counterexample, precisely, a~$G$-graded ring$R$ such that $Q^{gr}_{cl}(R)=R$ is gr-Noetherian but not gr-Artinian.

Criteria on group G for Goldie theorems to be true for all G-graded rings