Rings satisfying the radical formula
\(R\) will denote a commutative ring with identity and every module is unitary. By \(B<A\) we mean that \(B\) is a proper submodule of \(A\). For the submodules \(B\) and \(C\) of \(A\), we let \((B:C) =\{r\in R:rC \subseteq B\}\). If \(P\) is a proper submodule of \(A\) such that \(ra \in P\), \(r\in R\), \(a\in A\) implies either \(a\in P\) or \(r\in (P:A)\), then \(P\) is said to be prime in \(A\). In case \(A=R\), prime submodules coincide with prime ideals. The intersection of all prime submodules of \(A\) containing \(B\) is denoted by \(\text{rad} B\). For an ideal \(I\) of \(R\) let \(\sqrt I=\{r\in R:r^n\in I\), for some \(n\in\mathbb{N}\}\). The envelope of \(B\), \(E(B)\), is the set of all \(x \in A\) for which there exist \(r\in R\), \(a\in A\) such that \(x=ra\) and \(r^na\in B\) for some \(n\in \mathbb{N}\). We say that \(A\) satisfies the radical formula if for very \(B<A\), \(\text{rad} B= \langle E(B)\rangle\). A ring \(R\) satisfies the radical formula provided that if \(A\) is any \(R\)-module, then \(A\) satisfies the radical formula.
scientific article; zbMATH DE number 897176