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

Abstract

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 a gr-semisimple graded right classical quotient ring. The criterion for gr-semiprime rings is that the group $G$ is periodic. Actually, the sufficiency of periodicity was proved by the author in 2011 and the necessity of it follows from the well-known counterexample (1979). The main result of the paper concerns the gr-prime case. In this case, Goodearl and Stafford proved the graded 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: $(\forall g,h\in G)(\exists n\in \mathbb{N})(gh^n=h^ng)$. 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.