Nil modules and the envelope of a submodule

David Ssevviiri; Annet Kyomuhangi

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arXiv · 2408.16240

Nil modules and the envelope of a submodule

David Ssevviiri·Annet Kyomuhangi

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Abstract

Let $R$ be a commutative unital ring and $N$ be a submodule of an $R$-module $M$. The submodule $\langle E_M(N)\rangle$ 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, $\langle E_M(0)\rangle$ 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.

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David Ssevviiri, Annet Kyomuhangi. 2025-06-25. Nil modules and the envelope of a submodule. https://arxiv.org/abs/2408.16240

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