Weakly prime modules.
All rings are associative with identity and all modules are unitary left \(R\)-modules. The authors introduce the notion of weakly prime module as a module \(M\) with the property that the annihilator of any non-zero submodule of \(M\) is a prime ideal. A proper submodule \(P\) of \(M\) is called a weakly prime submodule if \(M/P\) is a weakly prime module. A module \(M\) is called compatible if its weakly prime submodules and prime submodules coincide. The authors show that every multiplication module is compatible and that every module is compatible if and only if every prime ideal is maximal if and only if the module \(R\oplus R\) is compatible. They characterize, over a commutative ring \(R\), the modules having each proper (non-zero) submodule weakly prime. Also, over a ring with finite classical Krull dimension, it is proved that every module has a prime submodule if and only if it has a weakly prime submodule.
scientific article; zbMATH DE number 2166171