Nil modules and the envelope of a submodule

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.

10 pages