SOME RESULTS ON STRONGLY PRIME SUBMODULES

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called strongly prime submodule if (P +Rx : M)y  P for x; y 2 M, implies that x 2 P or y 2 P. It is shown that a nitely generated R-module M is Artinian if and only if M is Noetherian and every strongly prime submodule of M is maximal. We also study the strongly dimension of a module which is dened to be the length of a longest chain of strongly prime submodules.