On Φ-powerful submodules and Φ-strongly prime submodules - Waheed Ahmad Khan (AIMS Mathematics, 2021)
Let R be a commutative ring with identity and N be a submodule of an R-module M. We say a nonnil submodule N of an R-module M is a Φ-powerful (resp., Φ-strongly prime) submodule, if Φ(N) is a powerful (resp., strongly prime) submodule of a module Φ(M). We show that a nonnil prime submodule N of an R-module M is a Φ-powerful submodule if and only if it is a Φ-strongly prime submodule. Similarly, if every prime submodule of an R-module M is a Φ-strongly prime, then we call it a Φ-pseudo-valuation module (Φ-PVM). We also prove that a faithful multiplication R-module M is Φ-PVM if and only if some maximal nonnil submodules of M are Φ-powerful. In this perspective, we analyze that M is Φ-PVM if and only if R is a PVD. In due course, we provide some characterizations of these submodules along with their relationships under certain conditions.