One dimensional domains which satisfy the radical formula are Dedekind domains
Let \(M\) be an \(R\)-module, \(R\) a commutative ring with 1. Let \(N\) be a submodule of \(M\) with \(N\neq M\). The envelope of \(N\) in \(M\), \(E_M (N)\), is defined to be the set \(\{rm:r\in R\) and \(m\in M\) such that \(r^nm\in N\) for some natural number \(n\geq 1\}\). It is clear that \(\langle E_M(N)\rangle\), the submodule generated by \(E_M(N)\), is contained in \(M\text{-rad}_RN\), the radical of \(N\) in \(M\). We say \(N\) satisfies the radical formula in \(M\), if \(M\text{-rad}_RN=\langle E_M(N)\rangle\). It was proved by Jenkins and Smith that any Dedekind domain satisfies the radical formula. The aim of this paper is to show the converse is true for one-dimensional Noetherian domains.
scientific article; zbMATH DE number 938537