International Journal of Analysis and Applications | 2026
Authors: Ali O.K.; Shadher A.H.; Shahoodh M.K.
DOI: 10.28924/2291-8639-24-2026-37
Journal: International Journal of Analysis and Applications
Year: 2026
Publisher: Etamaths Publishing
Document Type: Article
Open Access: All Open Access; Gold Open Access
Cited by: 0
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 A of ₳ is called PT-essential in ₳ (A≤PTE₳) in case there exist a submodule D of prime submodule P of ₳ in which A≰Τ,∩D≤Τ implies D≤Τ. Furthermore, let ₳ be an R-module and Τ≤₳. The submodule Aof ₳ is said to be P-Small T-essential in ₳ (A≤PST₳) in condition that any small submodule D of prime submodule Pof ₳ in which A≰Τ,∩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. © 2026 the author(s).
P-essential submodule; Prime submodule; Small submodule; T-essential submodule