The Probability that Ideals in a Number Ring are k-wise Relatively r-Prime

TL;DR

This paper derives an exact formula for the probability that n ideals in a number ring are k-wise relatively r-prime, extending understanding of ideal primality relationships in algebraic number theory.

Contribution

It introduces a precise formula for the probability of k-wise relative r-primality among ideals in a fixed number ring, a novel result in algebraic number theory.

Findings

Provides an exact probability formula for k-wise relatively r-prime ideals.

Extends the theoretical understanding of ideal primality in algebraic number rings.

Offers tools for further probabilistic analysis in algebraic number theory.

Abstract

We say that n ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime if any k of them are relatively r-prime. In this article, we provide an exact formula for the probability that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime.