One dimensional local domains and radical formula
Let \(R\) be a commutative ring with identity and \(M\) an \(R\)-module. Let \(P\) and \(N\) be proper submodules of \(M\). We say that \(P\) is a prime submodule whenever \(re\in P\) for \(r\in R\) and \(e\in M\), we have \(r\in (P:_RM)\) or \(e\in P\). The intersection of all prime submodules of \(M\) containing \(N\) is denoted by \(\text{rad}(N)\). Also, the envelope of \(N\) is denoted by \(E(N)\) and defined to be the set \[ \{x\in M|\, x=ay, a^ny\in N\, \text{for some}\, a\in R, y\in M\, \text{and}\, n\in\mathbb{N}\}. \] We say that \(M\) satisfies the radical formula if for every submodule \(L\) of \(M\), \(\text{rad}(L)=\langle E(L) \rangle\); and we say that \(R\) satisfies the radical formula if every \(R\)-module satisfies the radical formula. In this paper, the author shows that a local integral domain \((R, \mathfrak m)\) of dimension one satisfies the radical formula if and only if for all integers \(n\geq2\), and for every \(a_1, \dots, a_n\in R\) with \(a_i\not \in Ra_j\) for all \(i\neq j\), \[ (Ra_1:Ra_2+\dots+Ra_n)=\mathfrak m((Ra_1:Ra_2+\dots+Ra_n). \]
scientific article; zbMATH DE number 6495596