he concept of small monoform module was introduced by Hadi and Marhun, where a module U is called small monoform if for each non-zero submodule V of U and for every non-zero homomorphism f ∈ Hom R (V, U), implies that ker f is small submodule of V. In this paper the author dualizes this concept; she calls it co-small monoform module. Many fundamental properties of co-small monoform module are given. Partial characterization of co-small monoform module is established. Also, the author dualizes the concept of small quasi-Dedekind modules which given by Hadi and Ghawi. She show that co-small monoform is contained properly in the class of the dual of small quasi-Dedekind modules. Furthermore, some subclasses of co-small monoform are investiga

The main goal of this paper is to introduce and studya new concept named Ᵹ*-supplemented which can be considered as a generalization of W-supplemented modules and Ᵹ-hollow module. Also, we introduce a Ᵹ*-supplement submodule.Many relationshipsof Ᵹ*-supplemented modules are studied. Especially, we give characterizations of Ᵹ*-supplemented modules and relationship between this kind of modules and other kind modules for example every Ᵹ-hollow (Ᵹ-local) module is Ᵹ*-supplemented and by an example we show that the converse is not true.

The concept of fully pseudo stable Banach Algebra-module (Banach A-module) which is the generalization of fully stable Banach A-module has been introduced. In this paper we study some properties of fully stable Banach A-module and another characterization of fully pseudo stable Banach A-module has been given.

In this paper, the concept of semi-?-open set will be used to define a new kind of strongly connectedness on a topological subspace namely "semi-?-connectedness". Moreover, we prove that semi-?-connectedness property is a topological property and give an example to show that semi-?-connectedness property is not a hereditary property. Also, we prove thate semi-?-irresolute image of a semi-?-connected space is a semi-?-connected space.

      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.

In this work the concept of multiplicatively closed set of S-act have been introduced. The relation between multiplicatively closed subset of S-act and compactly packed of S-act have been studied and proved some properties of this concepts. Let U be a M. C. set of a monoid S and let U* be a U-closed subset of M. Ϣ is a subact of M which is maximal in M-U*. If [Ϣ:M] is maximal in S, then Ϣ is a prime subact of M.

Let R be a ring and let M be a left R-module. In this paper introduce a small pointwise M-projective module as generalization of small M- projective module, also introduce the notation of small pointwise projective cover and study their basic properties.
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Throughout this paper, T is a ring with identity and F is a unitary left module over T. This paper study the relation between semihollow-lifting modules and semiprojective covers. proposition 5 shows that If T is semihollow-lifting, then every semilocal T-module has semiprojective cover. Also, give a condition under which a quotient of a semihollow-lifting module having a semiprojective cover. proposition 2 shows that if K is a projective module. K is semihollow-lifting if and only if For every submodule A of K with K/( A) is hollow, then K/( A) has a semiprojective cover.