Modules whose certain submodules are prime.
Throughout \(R\) is a commutative ring with identity and all modules are unital \(R\)-modules. A non-zero module \(M\) is called prime if its zero submodule is prime, i.e., if \(x\in M\) and \(r\in R\) with \(rx=0\), then \(x=0\) or \(rM=0\). A proper submodule \(N\) of a module \(M\) is called a prime submodule of \(M\) if \(M/N\) is a prime module. A module \(M\) is called fully prime if every proper submodule is a prime submodule. The authors show that a module \(M\) is fully prime if and only if \(M\) is homogeneous semisimple. Modules in which non-zero direct summands are prime are also characterized. For a one-dimensional Noetherian domain \(R\), the modules in which the zero submodule is the only prime submodule are determined.
scientific article; zbMATH DE number 2166020