On prime submodules (Q2477989)

Throughout \(R\) is a commutative ring with identity. A non-zero module \(M\) is called secondary if for all \(x\in R\), either \(xM=M\) or there exists \(n\in \mathbb{N}\) such that \(x^nM=0\). The authors establish results on secondary modules and on the radicals of a submodule. Thus, in the first part of the paper they deal with the question as to when a representable module satisfies the radical formula. In the final part, they prove that if \(R/\text{rad}\,R\) is semisimple for any ring \(R\), then \(R\) satisfies the radical formula.