Universal gradings of orders. H W Lenstra, J; Silverberg, A

An order is a commutative ring of which the additive group is a finitely generated free abelian group, and a graded order is an order that is provided with a grading by some abelian group. Examples 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. Our main result is that every reduced order has a grading that is universal in a natural sense. Most of our proofs depend on the observation that the additive group of any reduced order can in a natural way be equipped with a lattice structure.

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