Tayyebeh Amouzegar,
Volume 8, Issue 3 (Vol. 8,No. 3, 2022)
Abstract
Introduction
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(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 (weak
τ-supplement) provided there exists some U⊆ Msuch thatM=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+YX≪MX andX+YY≪MY.
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-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 methods
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.
Results and discussion
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 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.
Conclusion
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-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.
