The extension dimension of group graded rings

TL;DR

This paper introduces the graded extension dimension for group graded rings, showing its relation to non-graded extension dimension in strongly graded cases and its invariance under certain equivalences.

Contribution

It defines the graded extension dimension and proves its equivalence to non-graded extension dimension for strongly graded rings, also exploring invariance properties.

Findings

Graded extension dimension equals non-graded extension dimension for strongly graded rings.

Graded and graded separable equivalences preserve extension dimension.

Provides new insights into the structure of group graded rings.

Abstract

In this paper, we introduce the concept of graded extension dimension for a group graded ring R, denoted by gr.ext.dim(R). We prove that when R is strongly graded, its graded extension dimension coincides with the non-graded extension dimension of both R itself and its degree-zero subring Re. Furthermore, we demonstrate that graded equivalence and graded separable equivalence preserve the extension dimension under appropriate conditions.