W-visible submodules and W-fully visible modules

In this paper, we presented a special type of submodule named W-visible submodule, which is weakerthan the visible submodule, where a proper submodule W of a T-module X is said to be W-visible if W = IW for somenon-zero proper ideal I of T. Also, we presented the concept of W-fully visible module, where a module X over aring T is said to be W-fully visible if any proper submodule of X is W-visible submodule. This concept is consideredweaker than the concept of fully visible module. In this study, we achieved numerous results and characteristics thatbelong to this concept.Keywords:Visible submodule; W-visible submodule; Fully visible module; W-fully visible module.1. INTRODUCTIONLetXbe a unitary module andTbe a commutative ring. In [1], Mahmood and Buthyna submitted a concept of visiblesubmodule which is defined as a proper submoduleWof a moduleXover a ringTto achieveW=IWfor every non-zeroidealIofT. Also, Mahmood [2] presented the concept of fully visible module, which is defined as the module in whichall submodules are visible.In section two of this paper, a special type of submodule that is weaker than the visible submodule was presented; thissubmodule has been called theW-visible submodule. In this submodule, a proper submoduleWof aT-moduleXis saidto beW-visible ifW=IWfor some non-zero proper idealIofT.It is clear that every visible submodule isW-visible, but the converse is not true, and we have been able to give anexample of this.In section three, we presented the concept ofW-fully visible module, which is a module filled withW-visible submod-ules; in other words, a moduleXover a ringTis said to beW-fully visible if any proper submodule ofXisW-visiblesubmodule. This concept is considered weaker than the concept of fully visible module.Many characterizations ofW-visible submodules andW-fully visible modules have been submitted to many differentproperties.