SOME RESULTS ON STRONGLY PRIMESUBMODULES

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

Keywords:

Classical Krull dimension , Strongly prime submodule