This thesis has three aims, The first aim of this thesis is to develop pure closed submodules, introduced by Nada M. Al-Thani in 1997. A submodule 𝒱 of an ℛ-module ℳ is termed 𝒫-essential if, for every pure submodule ℒ of ℳ with 𝒱⋂ℒ=0, then ℒ=0, and 𝒱 is pure closed if 𝒱 has no proper 𝒫-essential extension in ℳ.

Subsequently, the second aim of this thesis is to explore two new concepts: 𝒫𝑢𝑐-extending and strongly 𝒫𝑢𝑐-extending modules, using pure closed and 𝒫-essential submodules. These concepts are generalizations of the extending module. A module ℳ is named 𝒫𝑢𝑐-extending if each pure closed submodule of ℳ is a direct summand, and ℳ is strongly 𝒫𝑢𝑐-extending if each submodule of ℳ is 𝒫-essential in a direct summand.

In the third aim of this thesis, 𝒫𝑢𝑐-extending and strongly 𝒫𝑢𝑐-extending modules were extended to include two new generalizations: purely closed-extending and strongly purely closed-extending modules, respectively. An ℛ-module ℳ is named purely closed-extending if each pure closed submodule of ℳ is a pure submodule, and ℳ is strongly purely closed-extending if each submodule is 𝒫-essential in a pure submodule of ℳ.

Excellent