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

TL;DR

This paper establishes criteria based on group properties for when all G-graded Goldie rings have a well-behaved graded classical quotient ring, extending known results and providing counterexamples for certain groups.

Contribution

It introduces new criteria on the group G ensuring the existence of graded classical quotient rings for G-graded Goldie rings, extending previous results to broader classes of groups.

Findings

Periodic groups are necessary for gr-semiprime rings to have graded classical quotient rings.

Extended the class of groups for which graded Goldie's Theorem holds beyond Abelian groups.

Constructed counterexamples for groups outside the specified class, showing the limits of the criteria.

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.