发表机构
University of Dodoma; Makerere University（多多马大学; 马凯雷雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对交换环上理想的反向族，引入 Phi 约化模与协约化模，证明广义挠与完备化函子由最大真成员决定，并建立刻画、闭包性质及 Greenlees-May 型伴随，为理想族上的广义挠理论提供模论框架。
AI 中文摘要
设 R 为交换环，Phi 为理想的反向族，其最大真成员为 L。我们引入并研究 Phi 约化模与 Phi 协约化模，推广了单一理想幂的相应概念。我们证明，在这些模类上，与 Phi 相关的广义挠函子与完备化函子由 L 决定。我们建立了这些模的刻画与闭包性质，并得到 Greenlees-May 型伴随及 Matlis-Greenlees-May 型刻画。当 Phi 为理想系时，我们进一步研究广义挠函子的根性，并在合适的 Serre 子范畴上导出相关的挠理论。这些结果为研究关于理想族的广义挠与完备化提供了模论框架。
英文摘要
Let $R$ be a commutative ring and $Φ$ an inverse family of ideals with greatest proper member $\mathcal{L}$. We introduce and study $Φ$-reduced and $Φ$-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and completion functors associated with $Φ$ are determined by $\mathcal{L}$. We establish characterizations and closure properties of these modules and obtain a Greenlees-May type adjunction and a Matlis-Greenlees-May type characterization. When $Φ$ is a system of ideals, we further investigate the radicality of the generalized torsion functor and derive associated torsion theories on suitable Serre subcategories. These results provide a module-theoretic framework for studying generalized torsion and completion with respect to families of ideals.
