Strong Hollowness in Commutative Rings

TL;DR

This paper investigates the structure and properties of strongly hollow and completely strongly hollow ideals in commutative rings, providing characterizations, restrictions, and connections to other ideal concepts without finiteness assumptions.

Contribution

It introduces new structural insights and characterizations of strongly hollow ideals, including their relation to extremal ideals, irreducibility, and GCD conditions in commutative rings.

Findings

Strongly hollow ideals not contained in the Jacobson radical are completely strongly hollow.

Characterizations of completely strongly hollow ideals via extremal ideals.

Restrictions on strongly hollow ideals in integral domains and principal ideal domains.

Abstract

In this paper we study strongly hollow ideals and completely strongly hollow ideals in commutative rings without finiteness assumptions. We establish basic structural properties, including maximality phenomena and permanence under quotients and surjective homomorphisms. We obtain several characterizations of completely strongly hollow ideals in terms of extremal ideals avoiding a given ideal, and we show that a strongly hollow ideal which is not contained in the Jacobson radical is necessarily completely strongly hollow. As applications, we derive strong restrictions in integral domains and consequences for principal ideal domains, including a discrete valuation ring criterion. We develop the connection between complete hollowness and complete irreducibility and obtain a correspondence between completely strongly hollow ideals and completely strongly irreducible ideals. Finally, we develop a condition related to greatest common divisors which is equivalent to strongly hollowness under mild finiteness conditions.