# Universal gradings of orders

## TL;DR

This paper introduces the concept of universal gradings for reduced orders in commutative rings, proving their existence and properties, with applications to cryptography and lattice theory.

## Contribution

It defines universal gradings for reduced orders, proves their existence as finite group gradings, and extends properties of group rings to graded orders.

## Key findings

- Every reduced order has a finite group universal grading.
- Examples include group rings of finite abelian groups over number field rings.
- Applications to cryptography and lattice structures are demonstrated.

## Abstract

For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order} has a universal grading, and this grading is by a {\em finite} group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.