Simplified radical formula in modules (Q2901912)

Let \(R\) be a commutative ring and \(M\) a unitary \(R\)-module. A proper submodule \(P\) of \(M\) is called prime, if whenever \(rm\in P\) for some \(r\in R\) and \(m\in M\), then either \(m\in P\) or \(r\in (P :_R M)\), where \((P :_R M)\) is the ideal \(\{a\in R | aM\subseteq P\}\). If \(P\) is a prime submodule of \(M\), then \(p:=(P :_R M)\) is a prime ideal of \(R\) and \(P\) is called a \(p\)-prime submodule. If \(B\) is a submodule of \(M\), then the intersection of all prime submodules of \(M\) containing \(B\) is called the radical of \(B\) and denoted by \(\mathrm{rad}_M(B)\). If \(M\) has no prime submodules containing \(B\), then we consider \(\mathrm{rad}(B) = M\).NEWLINENEWLINEIn the paper under review, the authors introduced the concepts of simplified radical formula for modules and rings. The \(R\)-module \(M\) is said to satisfy the simplified radical formula, if for every submodule \(B\) of \(M\) and each \(x\in \mathrm{rad}(M)\), \(x=rm+B\), where \(r\in R\), \(m\in M\), \(b\in B\) and \(r^km\in B\) for some positive integer \(k\). Also we say that the ring \(R\) satisfies the simplified radical formula, if every \(R\)-module satisfied the simplified radical formula. The authors among the other results prove that a Noetherian ring satisfies the simplified radical formula if and only if it is a ZPI-ring. They also show that every valuation domain of Krull dimension \(1\) satisfies the simplified radical formula. Finally they characterize the zero dimensional local rings satisfying simplified radical formula.