مدول های گلدی مکمل پذیر در ارتباط با پیش رادیکال

 فرض کنید τ یک پیش رادیکال باشد. در این مقاله رابطه روی زیرمدول های یک مدول را تعریف و بررسی می کنیم. نشان می دهیم که رابطه βτ* یک رابطه هم ارزی است. این رابطه را برای تعریف مدول های گلدی τ-مکمل پذیر و مدول های به طور قوی   τ -h‎ مکمل پذیر  و  بررسی ویژگی های آنها  بکار می بریم.

goldie supplemented modules with respect to a preradical

introductionthroughout this paper r will denote an associative ring with identity, m a unitaryright r-module. a functor τfrom the category of the right r-modules mod-r to itself is called a preradicalif it satisfies the following properties:(i) τ(m)is a submodule of m, for every r-module m;(ii) if f:m’→mis an r-module homomorphism, then f(τm’≤τm and τ(f) is the restriction of fto τm’.for example rad, soc, and zmare preradicals. note that if k is a summand of m,then k∩τ(m)=τ(k).for a preradical  τ, al-takhman, lomp and wisbauer defined and studied the concept of τ-lifting and τ-supplemented modules. a module m is called τ-lifting if every submodule n of m has a decomposition n =a⊕ b such that a is a direct summand of m and b⊆τ(m).a submodule k⊆ m is called τ-supplement (weakτ-supplement) provided there exists some u⊆ msuch thatm=u+k andu∩ k⊆τ(k)    (u∩ k⊆τ(m)).m is called τ-supplemented (weakly τ-supplemented) if each of its submodulesτ-supplement (weak τ-supplement) in m.talebi, moniri hamzekolaei and keskin-tütüncü, defined τ-h-supplemented modules.  a module m is calledτ-h-supplemented if for every n≤ m there exists a direct summand d of msuch that(n+d)/n     ⊆τ(m/n)and(n+d)/d⊆τ(m/d).the β* relation is introduced and investigated by birkenmeier, takil mutlu, nebiyev, sokmez and tercan.  let x and y be submodules of m. x and yare β* equivalent,xβ*y, provided x+yx≪mx andx+yy≪my.based on definition of β* relation they introduced two new classes of modules namelygoldie*-lifting and goldie*-supplemented.they showed that two concept ofh-supplemented modules and goldie*-lifting modules coincide.in this paper, we introduce goldie-τ-supplemented and strongly τ-h-supplemented modules. we introduce the β* relation. we investigate some properties of this relation and prove that this relation is an equivalence relation. we define goldie-τ-supplemented and strongly τ-h-supplemented modules. we call a module m, goldie-τ-supplemented (strongly τ-h-supplemented) if for any submodule n of m,there exists a τ-supplement submodule (a direct summand) d of m such thatnβ*d. clearly every strongly τ-h-supplemented module is goldie τ -supplemented. we will study direct sums of goldie τ -h-supplemented modules. let m = a⊕ b be a distributive module. then m is goldie τ -upplemented(strongly τ -h-supplemented) if and only if a and b are goldie τ -supplemented(strongly τ -h-supplemented. we also define τ -h-cofinitely supplemented modules and obtain some conditions which under the factor module of a τ -h-cofinitely supplemented module will be τ -h-cofinitely supplemented.material and methodsin this paper, first we define and investigate the βτ*  relation on submodules of a module.  we show that the βτ*relation is an equivalence relation. we apply this relation to define and investigate the classes of goldie-τ -supplemented modules and stronglyτ-h-supplemented modules.results and discussionwe investigate some properties of this relation and prove that this relation is an equivalence relation. we define goldie-τ-supplemented and strongly τ-h- supplemented modules. we call a module m, goldie-τ-supplemented (strongly τ -h-supplemented) if for any submodule n of m, there exists a τ-supplement submodule (a direct summand) d of m such that nβ* d. clearly every strongly τ -h-supplemented module is goldie τ -supplemented. we will study direct sums of goldie τ -h-supplemented modules. let m = a⊕ b be a distributive module. then m is goldie τ -upplemented (strongly τ -h-supplemented) if and only if a and b are goldie τ -supplemented (strongly τ -h-supplemented). we also define τ -h-cofinitely supplemented modules and obtain some conditions which under the factor module of a τ -h-cofinitely supplemented module will be τ -h-cofinitely supplemented.conclusionthe following conclusions were drawn from this research.let m = m1⊕ m2, where m1 is a fully invariant submodule of m. assume that τ is a cohereditary preradical. if m is strongly τ-h-supplemented, then m1 and m2 are strongly τ-h-supplemented.let m be an τ-h-cofinitely supplemented module and let n≤ m be a submodule. suppose that for every direct summand k of m, there exists a submodule l of m such that n⊆ l⊆ k+n, l/n is a direct summand of m/n andk+nnl/n⊆τmn+lnl/n. then m/n is τ-h-cofinitelysupplemented.let m be a module and let n≤ m be a submodule such that for each decomposition m = m1⊕ m2 we have n = n∩ m1⊕ (n∩ m2). if m is τ-h-cofinitely supplemented, then m/n is τ-h-cofinitely supplemented.