On $ \Phi $-powerful submodules and $ \mathrm{\Phi} $-strongly prime submodules

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 $ \mathrm{\Phi} $-powerful (resp., $ \mathrm{\Phi} $-strongly prime) submodule, if $ \mathrm{\Phi}(N) $ is a powerful (resp., strongly prime) submodule of a module $ \mathrm{\Phi}(M) $. We show that a nonnil prime submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful submodule if and only if it is a $ \mathrm{\Phi} $-strongly prime submodule. Similarly, if every prime submodule of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-strongly prime, then we call it a $ \mathrm{\Phi} $-pseudo-valuation module ($ \mathrm{\Phi} $-PVM). We also prove that a faithful multiplication $ R $-module $ M $ is $ \mathrm{\Phi} $-PVM if and only if some maximal nonnil submodules of $ M $ are $ \mathrm{\Phi} $-powerful. In this perspective, we analyze that $ M $ is $ \mathrm{\Phi} $-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.