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; 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module; and lastly, 4) we construct a sheaf of nil $R$-modules on $\text{Spec}(R)$.
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无模块和子模块包络
让 $R$ 是交换单素环,$N$ 是 $R$ 模块$M$ 的子模块。由 $N$ 的包络 $E_M(N)$ 产生的子模块 $/langle E_M(N)\rangle$ 在研究满足激元公式的环和模块时非常重要。我们证明1) 半根是所有子模块的不变量,这些子模块分别由给定子模块的升序包络链中的包络生成;2) 对于满足根式的环,$\langle E_M(0)\rangle$ 是一个幂等根,它诱导扭转理论,其扭转类包括所有零$R$模块,无扭转类包括所有还原$R$模块;最后,4)我们在$text{Spec}(R)$上构造了一个零$R$模块的 Sheaf。
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