Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

Let R be a ring and let A be a unitary left R-module. A proper submodule H of an R-module A is called 2-absorbing , if rsa∈H, where r,s∈R,a∈A, implies that either ra∈H or sa∈H or rs∈[H:A], and a proper submodule H of an R-module A is called quasi-prime , if rsa∈H, where r,s∈R,a∈A, implies that either ra∈H or sa∈H. This led us to introduce the concept pseudo quasi-2-absorbing submodule, as a generalization of both concepts above, where a proper submodule H of an R-module A is called a pseudo quasi-2-absorbing submodule of A, if whenever rsta∈H,where r,s,t∈R,a∈A, implies that either rsa∈H+soc(A) or sta∈H+soc(A) or rta∈H+soc(A), where soc(A) is socal of an

In the geotechnical engineering applications, precise understandings are yet to be established on the effects of a foundation stiffness on its bearing capacity and settlement. The modern foundation construction uses the new available construction materials that totally change the relative stiffness of the footing structures-soil interactions such as waste material and landfill area of more residential purposes. Conventional bearing capacity equations were dealt with common rigid footing and thus cannot be used for reduced foundation rigidity. Therefore, this study investigates the effects of foundation relative stiffness on its load-displacement behaviour and the soil deformation field using compression test of a strip smooth footings on su

Abstract The relative pronoun in Hebrew language is an important pronoun use anciently and recently, it developed and it's usage and meanings differed so, it was not confined to the particle "אֲשֶׁר" as a relative pronoun, but beside it appeared other pronouns giving the relative meaning. Hence, the topic of this research was on this basis through studying the relative pronouns in old and modern Hebrew, the way of using them and their connection with preposition particles, as well as studying the relative clause.

In this paper, we present the almost approximately nearly quasi compactly packed (submodules) modules as an application of the almost approximately nearly quasiprime submodule. We give some examples, remarks, and properties of this concept. Also, as the strong form of this concept, we introduce the strongly, almost approximately nearly quasi compactly packed (submodules) modules. Moreover, we present the definitions of almost approximately nearly quasiprime radical submodules and almost approximately nearly quasiprime radical submodules and give some basic properties of these concepts that will be needed in section four of this research. We study these two concepts extensively.

In this paper, the concept of fully stable Banach Algebra modules relative to an ideal has been introduced. Let A be an algebra, X is called fully stable Banach A-module relative to ideal K of A, if for every submodule Y of X and for each multiplier ?:Y?X such that ?(Y)?Y+KX. Their properties and other characterizations for this concept have been studied.

     Let Ḿ be a unitary R-module and R is a commutative ring with identity. Our aim in this paper  to study the concepts T-ABSO fuzzy ideals, T-ABSO fuzzy submodules and T-ABSO quasi primary fuzzy submodules, also we discuss these concepts in the class of multiplication fuzzy modules and relationships between these concepts. Many new basic properties and characterizations on these concepts are given.

Single mode-no core-single mode fiber structure with a section of tuned no-core fiber diameter to sense changes in relative humidity has been experimentally demonstrated. The sensor performance with tuned NCF diameter was investigated to maximize the evanescent fields. Different tuned diameters of of (100, 80, and 60)μm were obtained by chemical etching process based on hydrofluoric acid immersion. The highest wavelength sensitivity was obtained 184.57 pm/RH% in the RH range of 30% –100% when the no-core fiber diameter diameter was 60 μm and the sensor response was in real-time measurements