Abstract
Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and introduce properties of this types of modules . Also we discuss the relation between them with examples and remarks are needed in our work.
Highlights
Throughout this paper, all rings are associative with unity and modules are unital left R–modules, where R denotes such a ring and M denotes such a module
A submodule N of M is called a small submodule of M if whenever N+ K= M for some submodule K of M, we have M = K, and in this case we write N
A nonzero module M is called hollow module, if every proper submodule of M is small in M
Summary
Throughout this paper , all rings are associative with unity and modules are unital left R–modules , where R denotes such a ring and M denotes such a module. Proposition (2.2) : Let A , B be submodule of an R–module M if A, B M and J( ) Proof : Let to prove has J–supplement in , K M , and M is J–supplemented , there exists L M such that M = K + L , and K L L , =
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