Goldie supplemented modules with respect to a preradical

Throughout this paper R will denote an associative ring with identity, M a unitary right 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(𝜏(𝑀′ )) ≤ 𝜏(𝑀) and 𝜏(𝑓) is the restriction of 𝑓to 𝜏(𝑀′ ). 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 that 𝑀 = 𝑈 + 𝐾 and 𝑈 ∩ 𝐾 ⊆ 𝜏(𝐾) (𝑈 ∩ 𝐾 ⊆ 𝜏(𝑀)). M is called 𝜏-supplemented (weakly 𝜏-supplemented) if each of its submodules 𝜏-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+𝑌 𝑋 ≪ 𝑀 𝑋 𝑎𝑛𝑑 𝑋+𝑌 𝑌 ≪ 𝑀 𝑌 . Based on definition of 𝛽 ∗ relation they introduced two new classes of modules namely 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ -lifting and 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ −supplemented.They showed that two concept of H-supplemented modules and 𝐺𝑜𝑙𝑑𝑖𝑒 ∗ −lifting modules coincide. In this paper, we introduce Goldie−𝜏 −supplemented and strongly 𝜏-Hsupplemented modules. We introduce the̅𝛽̅̅̅∗ 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 thatN𝛽̅̅̅∗D. Clearly every strongly 𝜏-H-supplemented module is Goldie 𝜏 -supplemented. We will study direct sums of Goldie 𝜏 -Hsupplemented 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 -Hcofinitely supplemented modules and obtain some conditions which under the factor module of a 𝜏 -H-cofinitely supplemented module will be 𝜏 -H-cofinitely supplemented.

In 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.

We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie−𝜏 −supplemented and strongly 𝜏-Hsupplemented 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 N𝛽̅̅̅∗ 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 -Hsupplemented) 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.

The 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-Hsupplemented, 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 and 𝐾+𝑁 𝑁 𝐿/𝑁 ⊆ 𝜏( 𝑀 𝑁 )+ 𝐿 𝑁 𝐿/𝑁 . Then M/N is 𝜏-Hcofinitelysupplemented.  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.