PT-Essential and PST-Essential Submodules

Abstract

Many generalizations have been presented in modules theory in order to present some new results in this direction. In this article, we introduced the concepts of PT-essential submodules and PST-essential submodules as a generalization to the concept of P-essential submodules. In particular, let ₳ be an R-module with Τ ≤ ₳. A submodule U of ₳ is called PT-essential in ₳ (U ≤_PTE ₳) in case there exist a submodule D of prime submodule P of ₳ in which U ≰ Τ, U ∩ D ≤ Τ implies D ≤ Τ. Furthermore, let ₳ be an R-module and Τ ≤ ₳. The submodule U of ₳ is said to be P-Small T-essential in ₳ (U ≤PST ₳) in condition that any small submodule D of prime submodule P of ₳ in which U ≰ Τ, U ∩ D ≤ Τ implies D ≤ Τ. Besides that, the concept of PT-complement submodule has been introduced with some properties about it. Finally, some basic properties of these concepts have been established. Then, several examples are given to illustrate the mentioned concepts.