Super-multiplicativity of ideal norms in number fields

TL;DR

This paper investigates the relationship between ideal norm inequalities and the minimal number of generators for ideals in subrings of number fields, establishing a key equivalence involving super-multiplicativity.

Contribution

It proves that in certain subrings of number fields, the ideal norm's super-multiplicativity characterizes ideals being generated by at most three elements.

Findings

Ideal can be generated by at most 3 elements in the specified subring.

Super-multiplicativity of ideal norms is equivalent to the ideal generation bound.

The result holds for all ring extensions within the normalization of the subring.

Abstract

In this article we study inequalities of ideal norms. We prove that in a subring $R$ of a number field every ideal can be generated by at most $3$ elements if and only if the ideal norm satisfies $N(IJ) \geq N(I)N(J)$ for every pair of non-zero ideals $I$ and $J$ of every ring extension of $R$ contained in the normalization of $R$.