Strong Hollowness in Commutative Rings
Amartya Goswami, Joseph Israel Zelezniak
TL;DR
The paper develops a comprehensive framework for strongly hollow and completely strongly hollow ideals in commutative rings without finiteness assumptions. It introduces $\Gamma_I$ and $L_I$ as key invariants, proves existence/maximality via Zorn’s lemma, and shows persistence under quotients and surjective maps, along with precise behavior of $\Gamma_I$ and $L_I$ under quotients. It provides two central characterizations of completely strongly hollow ideals and establishes a Jacobson-radical criterion: if a strongly hollow ideal is not contained in $J(R)$, it is completely strongly hollow, linking hollowness to radical containment and localization. A major contribution is the duality between completely strongly hollow and completely strongly irreducible ideals, with explicit bijections mediated by $\Gamma_I$ and $(\cdot:M)$, and its implications in Noetherian/Artinian settings. The gcd-based $(\star)$ condition in GCD rings yields practical criteria: finitely generated ideals satisfying $(\star)$ are strongly hollow, with partial converses in Bezout rings and extensions to ACCP contexts, enriching the structural theory of hollowness and its connection to divisibility properties.
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.