Containment-Division Rings and New Characterizations of Dedekind Domains

TL;DR

This paper introduces Containment-Division Rings (CDRs), a new class of rings, and demonstrates their equivalence to Dedekind domains in Noetherian cases, providing new characterizations and insights into ring theory.

Contribution

The paper defines CDRs, shows their connection to Dedekind domains, and introduces a weaker chain condition, expanding the understanding of ring classifications.

Findings

CDRs are essentially equivalent to Dedekind domains in Noetherian cases.

The Noetherian condition can be replaced by a Divisor Chain Condition for CDRs.

The concept of CDRs was co-discovered with computer assistance, highlighting novel methods in mathematical discovery.

Abstract

We introduce a new class of commutative rings with unity, namely, the Containment-Division Rings (CDR-s). We show that this notion has a very exceptional origin since it was essentially co-discovered with the qualitative help of a computer program (i.e. The Heterogeneous Tool Set (HETS)). Besides, we show that in a Noetherian setting, the CDR-s are just another way of describing Dedekind domains. Simultaneously, we see that for CDR-s, the Noetherian condition can be replaced by a weaker Divisor Chain Condition.