Strongly Essential Submodules and Modules with the se-CIP

Abstract

     Let  be a ring with identity. Recall that a submodule  of a left -module  is called strongly essential if for any nonzero subset  of , there is  such that , i.e., . This paper introduces a class of submodules called se-closed, where a submodule  of  is called se-closed if it has no proper strongly essential extensions inside . We show by an example that the intersection of two se-closed submodules may not be se-closed. We say that a module  is have the se-Closed Intersection Property, briefly se-CIP, if the intersection of every two se-closed submodules of  is again se-closed in . Several characterizations are introduced and studied for each of these concepts. We prove for submodules  and  of  that a module  has the se-CIP if and only if  is strongly essential in  implies  is strongly essential in . Also, we verify that, a module  has the se-CIP if and only if for each se-closed submodule  of  and for all submodule  of ,  is se-closed in . Finally, some connections and examples are included about (se-CIP)-modules