{"id":10735,"date":"2025-12-14T13:50:53","date_gmt":"2025-12-14T10:50:53","guid":{"rendered":"https:\/\/en.csw.uobaghdad.edu.iq\/?p=10735"},"modified":"2025-12-14T13:53:47","modified_gmt":"2025-12-14T10:53:47","slug":"a-thesis-in-the-college-of-science-for-women-discussing-some-generalizations-of-extending-modules-relative-to-pure-closed-and-%f0%9d%92%ab-essential-submodules","status":"publish","type":"post","link":"https:\/\/en.csw.uobaghdad.edu.iq\/?p=10735","title":{"rendered":": A thesis in the College of Science for Women discussing Some Generalizations of Extending Modules Relative to Pure closed and \ud835\udcab-essential Submodules"},"content":{"rendered":"

[vc_row full_width=”stretch_row” rtl_reverse=”yes” css=”.vc_custom_1726859899671{margin-top: -150px !important;padding-top: 150px !important;padding-bottom: 60px !important;background-color: #f5f5f5 !important;}”][vc_column width=”1\/2″]

<\/div>

Master’s <\/span><\/span>Discussion<\/span><\/span><\/p>\n

The College of Science for Women at the University of Baghdad discussed a master’s thesis<\/p>\n

(Some Generalizations of Extending Modules Relative to Pure closed and \ud835\udcab-essential Submodules)<\/p>\n

by a student (Alaa Abdul Azeez Abdul Ameer).<\/p>\n<\/div><\/div><\/div>

<\/div><\/div>
<\/div>
<\/div>
<\/i>
Objectives of the Thesis<\/div><\/span>
[vc_column_text css=””]This thesis has three aims, The first aim of this thesis is to develop pure closed submodules, introduced by Nada M. Al-Thani in 1997. A submodule \ud835\udcb1 of an \u211b-module \u2133 is termed \ud835\udcab-essential if, for every pure submodule \u2112 of \u2133 with \ud835\udcb1\u22c2\u2112=0, then \u2112=0, and \ud835\udcb1 is pure closed if \ud835\udcb1 has no proper \ud835\udcab-essential extension in \u2133.<\/p>\n

Subsequently, the second aim of this thesis is to explore two new concepts: \ud835\udcab\ud835\udc62\ud835\udc50-extending and strongly \ud835\udcab\ud835\udc62\ud835\udc50-extending modules, using pure closed and \ud835\udcab-essential submodules. These concepts are generalizations of the extending module. A module \u2133 is named \ud835\udcab\ud835\udc62\ud835\udc50-extending if each pure closed submodule of \u2133 is a direct summand, and \u2133 is strongly \ud835\udcab\ud835\udc62\ud835\udc50-extending if each submodule of \u2133 is \ud835\udcab-essential in a direct summand.<\/p>\n

In the third aim of this thesis, \ud835\udcab\ud835\udc62\ud835\udc50-extending and strongly \ud835\udcab\ud835\udc62\ud835\udc50-extending modules were extended to include two new generalizations: purely closed-extending and strongly purely closed-extending modules, respectively. An \u211b-module \u2133 is named purely closed-extending if each pure closed submodule of \u2133 is a pure submodule, and \u2133 is strongly purely closed-extending if each submodule is \ud835\udcab-essential in a pure submodule of \u2133.[\/vc_column_text]<\/div><\/div>

<\/i>
Thesis Grade<\/div><\/span>
[vc_column_text css=””]Excellent <\/strong>[\/vc_column_text]<\/div><\/div>
<\/i>
Picture of the Committee members<\/div><\/span>
[vc_column_text css=””]<\/p>\n
\n

 <\/p>\n

\"\"<\/div>\n<\/div>\n

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<\/div>[vc_single_image image=”10736″ img_size=”700×700″ alignment=”right” css=””][\/vc_column][\/vc_row]<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

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