On Large-Small submodule and Large-Hollow module

Abstract

The goal of this research is to introduce the concepts of Large-small submodule and Large-hollow module and some properties of them are considered, such that a proper submodule N of an R-module M is said to be Large-small submodule, if N + K = M where K be a submodule of M, then K is essential submodule of M ( K ≤e M ). An R-module M is called Large-hollow module if every proper submodule of M is Large-small submodule in M.

δ-Small submodule and Lifting property

“…A T-module M is called an S-L-hollow module if M has a unique maximal submodule that contains each S-small submodule of M [1].…”

Soc–L-Small –T–ABSO Submodules and Related Concepts

“…Assume that V be a ring with identity and W be a unitary left-module. A. Yousefian and F. Soheilnia in 2011 introduce the concept of T-Absorbing and weakly T-Absorbing submodules where " A proper submodule A of an V − module W is said to be T-ABSO submodule of W if whenever a, b ∈ V, x ∈ W such that abx ∈ A, then either ax ∈ A or bx ∈ A or ab ∈ (A: V W) [3] [4 ]", and " A proper submodule A of V-module W, is called small two absorbing submodule (small-T-ABSO) submodule of M denoted if a, b ∈ V and x ∈ W with < x >≪ W and abx ∈ A then ax ∈ A or bx ∈ A or ab ∈ (A: V W) [2 ]",In [1] A.A.Abduljaleel present the idea of large-small submodule as an expansion of a small submodule so that "A proper submodule A of V-module W, is called Large small (L-small) submodule of M denoted by (A ≪ L W), if A+H=W where H a submodule of W,then H ≤ e W". [ 1] provided the notion of a Socle-Two-Absorbing submodule, in which " A proper submodule A of an V-module W is called Socle -Two-Absorbing (in short is Soc-T-ABSO submodule of W if wheneve a, b ∈ V, x ∈ W such that abx ∈ A , then either ax ∈ A + Soc(W) or bx ∈ A + Soc(W) or ab ∈ (A + Soc(W): V W) [5]".…”