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 approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.
The aim of this work is studying many concepts of a pure submodule related to sub-module L and introducing the two concepts, T_pure submodule related to submodule and the crossing property of T_pure related to submodule. Another characterizations and study some properties of this concept.
Let R be an individual left R-module of the same type as W, with W being a ring containing one. W’s submodules N and K should be referred to as N and K, respectively that K ⊆ N ⊆ W if N/K <<_J (D_j (W)+K)/K, Then K is known as the D J-coessential submodule of Nin W as K⊆_ (Rce) N. Coessential submodule is a generalization of this idea. These submodules have certain interesting qualities, such that if a certain condition is met, the homomorphic image of D J- N has a coessential submodule called D J-coessential submodule.
Through this paper R represent a commutative ring with identity and all R-modules are unitary left R-modules. In this work we consider a generalization of the class of essential submodules namely annihilator essential submodules. We study the relation between the submodule and his annihilator and we give some basic properties. Also we introduce the concept of annihilator uniform modules and annihilator maximal submodules.
Five species of Trogoderma Berthold are reported from Iraq. These are: I inclus,n LeCont. T.
granarium Everts, T. variable Ballion. I. bactrianum Zantiev and T. irroraturn Reitter. A key
to the species is presented, with redeseriptions and illustrations of all species.
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Sixteen species of Armored Scale insects were recorded from Baghdad city during 2001-2005. Three of these are reported here for the first time Abgrallaspis cyanophylli (Signoret, 1869), Aonidiella citrina (Craw,1870) and Chrysomphalus aonidium (Linnaeus,1758). The other thirteen species were recorded earlier Aonidiella aurantii (Maskell), Aonidiella orientalis (Newstead), Chrysomphalus dictyospermi (Morgan), Diaspidiotus ostreaeformis (Curtis), Diaspidiotu perniciosus (Comctock), Hemiberlesia lataniae (Signoret), Lepidosaphes beckii (Newman), Lepidosaphes conchiformis (Gmelin), Lepidosaphes ulmi (Linnaeus), Mercetaspis halli
... Show MoreThis work includes the geographical distribution variation and notes for the habitat of Iraqi blind snakes Leptotyphlops macrorhynehus. Typhlops braminus, and Typhlops vermicularis. A key was also given for the identification of these three snakes.