Supplemented Modules Relative to an Ideal

Abstract

Let I be an ideal of a ring R and let M be a left R-module. A submodule L of M is said to be δ-small in M provided M = L + X for any proper submodule X of M with M/X singular. An R-module M is called I-⊕-supplemented if for every submodule N of M , there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K is δ-small in K. In this paper, we investigate some properties of I-⊕-supplemented modules. We also compare I-⊕-supplemented modules with ⊕-supplemented modules. The structure of I-⊕-supplemented modules and ⊕-δ-supplemented modules over a Dedekind domain is completely determined.

I-Rad-⊕-supplemented modules

“…In this paper, we define I-Rad-⊕-supplemented modules which is specialized of Rad-⊕-supplemented modules. We obtain various properties of this modules adapting by [14]. We show that every finite direct sum of I-Rad-⊕-supplemented modules is a I-Rad-⊕-supplemented module.…”

Güçlü Direkt Radikal Tümlenmiş Modüllerin İki Yeni Genelleştirmesi

“…Hatırlatmak gerekirse bir 𝐴 modülü, 𝐴 = 𝐴 1 + 𝐴 2 koşulunu sağlayan 𝐴 1 , 𝐴 2 ≤ ⨁ 𝐴 için 𝐴 1 ∩ 𝐴 2 ≤ ⊕ 𝐴 oluyorsa 𝐴 modülü (𝐷3) özelliğini sağlıyor denir. İspat: Teo 2.7 [12]…”

Some Variations of δ -Supplemented Modules with Regard to a Hereditary Torsion Theory

“…we say X is a fully invariant submodule of W. We refer the interested readers to [4][5][6] for concepts given here. Now, we give place to fundamental concepts of torsion theory.…”