$z^\circ$-submodules of a reduced multiplication module

TL;DR

The paper introduces $z^ast{}$-submodules as an extension of $z^ast{}$-ideals and studies them within the framework of reduced multiplication $R$-modules. It develops a detailed structural and topological picture of minimal prime submodules via $O_P$, $V(x)$, and $V(Ann_R(x)M)$, and provides annihilator-based characterizations such as $N=\sum_{x\in N}\mathfrak{P}_x$. It further links these submodules to torsion and radical notions, e.g., $T_0(M)$ and $Rad_N(M)$, and offers transfinite constructions and quotient-compatibility results. Overall, the work generalizes $z^ullet{}$-ideals to a submodule setting, supplying criteria and representations to construct and recognize $z^ast{}$-submodules in reduced multiplication modules, with implications for the prime-spectrum topology of $M$.

Abstract

Let R be a commutative ring with identity and M be an R-module. A proper ideal I of R is said to be a $z^\circ$-ideal if for each $a \in I$ the intersection of all minimal prime ideals containing a is contained in I. The purpose of this paper is to introduce the notion of $z^\circ$-submodules of M as an extension of $z^\circ$-ideals of R. Moreover, we investigate some properties of this class of submodules when M is a reduced multiplication R-module.

Theorems & Definitions (53)

• Corollary 2.1
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 2.7