Annihilator-based dependency relations in modules and radical characterizations

Abstract

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.