The Center of Crystalline Graded Rings

TL;DR

This paper explores the structure of the center in crystalline graded rings, analyzing cases with Abelian and non-Abelian finite grading groups, and introduces the concept of Arithmetically Crystalline Graded rings.

Contribution

It provides a comprehensive description of the center of crystalline graded rings and introduces the new concept of Arithmetically Crystalline Graded rings for non-principal cases.

Findings

The center of crystalline graded rings has properties similar to the rings themselves.

The center is Arithmetically Crystalline Graded in general.

Conditions for a trivial center are identified in the non-Abelian case.

Abstract

In the first section of the paper, we will give some basic definitions and properties about Crystalline Graded Rings. In the following section we will provide a general description of the center. Afterwards, the case where the grading group is Abelian finite will be handled. The center will have some properties of a crystalline graded ring, but not all. We will call this Arithmetically Crystalline Graded. The center is crystalline graded if the part of degree zero is a principal ideal domain. The last section deals with the case where the grading group is non-Abelian finite. Since this situation is much more complicated than the Abelian case, we primarily focus on the conditions to have a trivial center. The fact that the center is Arithmetically Crystalline Graded also holds in this case.