Annihilator-based dependency relations in modules and radical characterizations
In this paper we introduce and investigate annihilator based dependency relations for submodules of a unitary left \(R\)-module \(M\) over a commutative Noetherian ring \(R\). We show that two submodules \(N_1, N_2 \leq M\) are radically dependent (in the sense that \(\sqrt{\text{Ann}(N_1+N_2)}=\sqr...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2026
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2471 |
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| Journal Title: | Algebra and Discrete Mathematics |
| Download file: |  |
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Algebra and Discrete Mathematics| Summary: | In this paper we introduce and investigate annihilator based dependency relations for submodules of a unitary left \(R\)-module \(M\) over a commutative Noetherian ring \(R\). We show that two submodules \(N_1, N_2 \leq M\) are radically dependent (in the sense that \(\sqrt{\text{Ann}(N_1+N_2)}=\sqrt{\text{Ann}(N_1)}+\sqrt{\text{Ann}(N_2)}\)) if and only if \(\sqrt{\text{Ann}(N_1)}=\sqrt{\text{Ann}(N_2)}\). Building on this characterization, we introduce totally annihilator-dependent modules via a Krull-dimension condition and prove that, for a finitely generated module over a Noetherian ring, total annihilator-dependence is equivalent to \(\text{Ass}(M)\) being a singleton. We further study the Radical Distinction Set \(Z_g(M)\), establish its connection to associated primes, and extend the main results to finitely generated multiplication modules. |
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| DOI: | 10.12958/adm2471 |