Essentially Retractable Modules Relative To A Submodule And Some Generalizations

Abstract

R is a ring with unity, and all modules are unitary right R-modules. The concept of compressible modules was introduced in 1981 by Zelmanowitz, where module M is called compressible if it can be embedded in any nonzero submodule A of M . In other words, M is a compressible module if for each nonzero submodule A of M, f 2 Hom(M;A) exists, such that f is monomorphism. Retractable modules were introduced in 1979 Khuri, where module M is retractable if Hom(M, A ) 6= 0 for every nonzero submodule A of M . We define a new notion, namely, essentially retractable module relative to a submodule. In addition, new generalizations of compressible modules relative to a submodule are introduced, where module M is called compressible module relative to a submodule N of M . If for all nonzero submodule K of M contains N , then a monomorphism f 2 Hom(M, K) exists. Some basic properties are studied and many relationships between these classes and other related concepts are presented and studied. We also introduce another generalization of retractable module, which is called small kernel retractable module

Study the Optical and Morphological Properties of (PMMA/PS/Al2O3) Nanocomposites Before and After Exposed to Argon Plasma

“…One of the primary advantages of utilizing a polymer blend is the ability to customize the characteristics of the end product to fulfill the demands of specific applications, a Feature that cannot be accomplished when utilizing a solitary polymer type [1,2]. Creating a nanocomposite involves combining discrete nanoscale constituents to regulate and improve the structure and properties of the material [3] Increased tensile strength, conductivity, and thermal stability, in addition to reduced flammability [4]. In this regards, PMMA is a widely recognized and innovative polymer.…”

“…Polystyrene (PS), a pliable plastic material, is extensively utilized in various industries and everyday activities due to its costeffectiveness, lightweight nature, ease of production, flexibility, thermal efficiency, durability, and moisture resistance [9,10] [4,11]. The compound of aluminum oxide (Al 2 O 3 ) has attarcted attention for its exceptional stability, ability to withstand harsh conditions, and impressive transparency that extends to 250 nm, as reported in literature [11].…”