Nil modules and the envelope of a submodule

2024/08/29 by David Ssevviiri, Ssevviiri, David, Annet Kyomuhangi +1
Mathematics · #13C13 #16S90 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2408.16240

openalex publication_date 2024/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a commutative unital ring and N be a submodule of an R-module M. The submodule ⟨ E_M(N)⟩ generated by the envelope E_M(N) of N is instrumental in studying rings and modules that satisfy the radical formula. We show that: 1) the semiprime radical is an invariant on all the submodules which are respectively generated by envelopes in the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, ⟨ E_M(0)⟩ is an idempotent radical and it induces a torsion theory whose torsion class consists of all nil R-modules and the torsionfree class consists of all reduced R-modules; and 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module.

Related

• Envelopes and Weakly Radicals of Submodules
• A new notion of semiprime submodules
• A complete radical formula and 2-primal modules
• Associated prime submodules of finitely generated modules
• Nilpotent elements control the structure of a module