Regular multiplication modules

On Strongly F – Regular Modules and Strongly Pure Intersection Property

“…All rings are commutative with identity element and all modules are unitary left modules, unless otherwise stated . Following [1] a submoduleA of a module is called strongly pure if for each finite subset{a i } in A , (equivalently , for each a A) there exists ahomomorphism f : M A such that f(a i ) =a i , i . M is Z regular if for each a M ,f M* = Hom (M , R) such that a = f (a) a .…”

“…M is Z regular if for each a M ,f M* = Hom (M , R) such that a = f (a) a . Equivalently , each f.g. submodule of M is projective direct summand [1] . M is Fregular if eachsubmodule of M is pure .…”

“…M is Fregular if eachsubmodule of M is pure . It is known that if N is a finitely generated strongly pure submodule of M , then N is a summand [1] . Clearly that every strongly pure submodule of a module M is pure ,The converse is true if M is projective [1] .…”

“…It is known that if N is a finitely generated strongly pure submodule of M , then N is a summand [1] . Clearly that every strongly pure submodule of a module M is pure ,The converse is true if M is projective [1] . Note that a ring R is Z-regular module iff R is strongly Fregular iff R is Fregular module iff R is a regular ring (in the sense of Von Neumann) [1] .…”

“…Clearly that every strongly pure submodule of a module M is pure ,The converse is true if M is projective [1] . Note that a ring R is Z-regular module iff R is strongly Fregular iff R is Fregular module iff R is a regular ring (in the sense of Von Neumann) [1] . Let R be an associative ring withidentity , and let M be a (left) unitary module .…”