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.

In this paper, we shall introduce a new kind of Perfect (or proper) Mappings, namely ω-Perfect Mappings, which are strictly weaker than perfect mappings. And the following are the main results: (a) Let f : X→Y be ω-perfect mapping of a space X onto a space Y, then X is compact (Lindeloff), if Y is so. (b) Let f : X→Y be ω-perfect mapping of a regular space X onto a space Y. then X is paracompact (strongly paracompact), if Y is so paracompact (strongly paracompact). (c) Let X be a compact space and Y be a p*-space then the projection p : X×Y→Y is a ω-perfect mapping. Hence, X×Y is compact (paracompact, strongly paracompact) if and only if Y is so.

      This paper investigates the concept (α, β) derivation on semiring and extend a few results of this map on prime semiring. We establish the commutativity of prime semiring and investigate when (α, β) derivation becomes zero.

تشغل الفضاءات الداخلية الطبية اهتمام واسع , لما توفره من رعاية صحية للمرضى ,فلابد ان يُهتَمْ بها من الجانب الوظيفي(الادائي), لتحقيق الراحة البصرية والنفسية والجسدية لغرض الوصول الى الاداء الجيد للكادر الطبي, ولهذا وجد ضرورة التعرف على تلك الفضاءات الداخلية بشكل اعمق , وهل انها ملاءمة للمرتكزات التصميمية المتعارف عليها؟ , لذلك تم تسليط الضوء على الفضاءات الداخلية للمختبرات الطبية, وقد تناول البحث المشكلة واه

In this paper, some basic notions and facts in the b-modular space similar to those in the modular spaces as a type of generalization are given.  For example, concepts of convergence, best approximate, uniformly convexity etc. And then, two results about relation between semi compactness and approximation are proved which are used to prove a theorem on the existence of best approximation for a semi-compact subset of b-modular space.