Annihilator Based Dependency Relations in Modules and Radical Characterizations

This paper introduces two novel concepts in module theory based on the structural behavior of annihilator ideals and their radicals: totally annihilator-dependent modules and a radical dependency relation between modules. These notions aim to generalize classical submodule dependence—such as linear or essential dependence—by incorporating the radical behavior of annihilator ideals into the algebraic analysis of modules over commutative Noetherian rings. Within the framework of radical dependency, denoted by $N_{1} \triangleleft_{rad} N_{2}$, we establish that two submodules are radically dependent if and only if their radical annihilators satisfy a mutual containment relation, specifically that either $\sqrt{Ann(N_{1})} \subseteq \sqrt{Ann(N_{2})}$ or $\sqrt{Ann(N_{2})} \subseteq \sqrt{Ann(N_{1})}$. We further provide a comprehensive characterization theorem for totally annihilator-dependent modules, proving that a finitely generated module $M$ satisfies this dependency condition if and only if its set of associated primes, $Ass(M)$, consists of a single prime ideal. Additionally, for finitely generated multiplication modules, we demonstrate that the radical of the annihilator of the sum of two nonzero submodules is equal to the sum of the radicals of their respective individual annihilators under the condition of a single associated prime. The study also explores the properties of radically supplemented modules and introduces the set $Z_{g}(M)$ to translate these dependency relations from submodules to module elements via annihilator radical equivalency classes. Several illustrative examples are provided to demonstrate the scope and limitations of these results, offering a new perspective for classifying modules through the lens of annihilator radicals and Krull dimension.