Strong Hollowness in Commutative Rings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911383841406976 |
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| author | Goswami, Amartya Zelezniak, Joseph Israel |
| author_facet | Goswami, Amartya Zelezniak, Joseph Israel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12388 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong Hollowness in Commutative Rings Goswami, Amartya Zelezniak, Joseph Israel Commutative Algebra Rings and Algebras 13A15, 13C13, 13F99 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. |
| title | Strong Hollowness in Commutative Rings |
| topic | Commutative Algebra Rings and Algebras 13A15, 13C13, 13F99 |
| url | https://arxiv.org/abs/2601.12388 |