Nil modules and the envelope of a submodule

TL;DR

This paper investigates the properties of envelopes of submodules in modules over commutative rings, revealing their role in radical theory, torsion theories, and specific module classes like Noetherian uniserial modules.

Contribution

It establishes the invariance of the semiprime radical on certain submodules, characterizes radicals induced by envelopes, and shows that Noetherian uniserial modules satisfy the semiprime radical formula.

Findings

Semiprime radical is invariant on submodules generated by envelopes.

Envelope-generated radical induces a torsion theory with nil and reduced modules.

Noetherian uniserial modules satisfy the semiprime radical formula.

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.