B-orderings for all ideals B of Dedekind domains and generalized factorials

2025/02/26 by Jeffrey C. Lagarias, Lagarias, Jeffrey C., Wijit Yangjit +1
Mathematics · #11A63 #11B65 #13F05 (Primary) #13F20 (Secondary) #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2502.19072

openalex publication_date 2025/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper extends Bhargava's theory of \mathfrakp-orderings of subsets S of a Dedekind ring R valid for prime ideals \mathfrakp in R. Bhargava's theory defines for integers k≥1 invariants of S, the generalized factorials [k]!_S, which are ideals of R. This paper defines \mathfrakb-orderings of subsets S of a Dedekind domain D for all nontrivial proper ideals \mathfrakb of D. It defines generalized integers [k]_S,T, as ideals of D, which depend on S and on a subset T of the proper ideals \mathscrI_D of D. It defines generalized factorials [k]!_S,T and generalized binomial coefficients, as ideals of D. The extension to all ideals applies to Bhargava's enhanced notions of r-removed \mathfrakp-orderings, and \mathfrakp-orderings of order h.

Related

• Simultaneous \mathfrakp-orderings and equidistribution
• Intrinsic factorization of ideals in Dedekind domains
• Factorization from an order-theoretic view 1&2
• Generalized Maximal Orders
• On Dedekind domains whose class groups are direct sums of cyclic groups