On modules which satisfy the radical formula (Q2839021)

Let \(R\) be a commutative ring with identity and \(M\) be a unitary \(R\)-module. A submodule \(N\) of \(M\) is is said to be a \textit{prime submodule} of \(M\), when for any \(r\in R\) and \(m\in M\), \(rm\in N\) implies that either \(m\in N\) or \(rM\subseteq N\). The (prime) radical of \(N\) in \(M\) is defined to be the intersection of all prime submodules of \(M\) containing \(N\) and is denoted by \(\text{rad}_M(N)\). The subset NEWLINE\[NEWLINEE_M(N)=\{rm : r \in R \text{ and } m\in M \text{ such that } r^km\in N \text{ for some } k\in N , k\geq 1\}NEWLINE\]NEWLINE is called the \textit{envelope } of \(N\) in \(M\). Obviousely \(N\subseteq RE_M(N)\subseteq\text{rad}_M(N)\) and the first inequality holds, precisely when \(N\) is a prime submodule. \(N\) is said to \textit{satisfy the radical formula} in \(M\), if the above bound holds.NEWLINENEWLINEIt is shown in [\textit{H. Sharif} et al., Acta Math. Hung. 71, No. 1--2, 103--108 (1996; Zbl 0890.13001)], that all submodules of \(M\) satisfy the radical formula, if \(R\) is an Artinian ring. The paper under review extends this result to the class of all representable modules.