aspg-21-3539-1791475363
20261008160243
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American Scientific Publishing Group
International Journal of Neutrosophic Science
IJNS
2692-6148
2690-6805
2025
25
4
On Modules Related to Homomorphism Their Kernel Equal Zero in Neutrosophic Theory
Firas N.
Hameed
General Directorate of Anbar Education, Ministry of Education, Ramadi, Anbar, Iraq
Fawzi N.
Hammad
Department of Mathematics, College of Education for Pure Sciences, University Of Anbar, Ramadi, Iraq
Majid Mohammed
Abed
Department of Mathematics, College of Education for Pure Sciences, University Of Anbar, Ramadi, Iraq
Neutrosophic set is a modern branch as a generalization of fuzzy concept. Zadeh in 1965 presented fuzzy concept and later he introduced more applications in more subjects of mathematics. On of the type branch of mathematics is fuzzy algebra. In this work, we present and clarify several results of several modules, which has zero-kernel, and zero homomorphism in neutrosophic theory. The aim modules are mnonoform and small monoform modules. Several concepts have been studied in this paper like Quasi-dedekind and uniform modules. We proved that if (
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"Times New Roman";mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;
mso-fareast-language:EN-US;mso-bidi-language:AR-IQ'>
m:val="p"/>M (
style='mso-bidi-font-style:normal'>
107%;font-family:"Cambria Math",serif;mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>T )) is a module over neutrosophic ring
style='mso-bidi-font-style:normal'>
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mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>
m:val="i"/>N (
style='mso-bidi-font-style:normal'>
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mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>T ). If
style='mso-bidi-font-style:normal'>
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mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
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EN-US;mso-bidi-language:AR-IQ'>M
m:val="script"/>(
style='mso-bidi-font-weight:normal'>
107%;font-family:"Cambria Math",serif;mso-fareast-font-family:Calibri;
mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>T ) is a directed sum of simple submodules an
style='mso-bidi-font-style:normal'>
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mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
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EN-US;mso-bidi-language:AR-IQ'>S
m:val="i"/>M(
m:val="script"/>T
m:val="i"/>) is monoform, then
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mso-fareast-font-family:Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:
"Times New Roman";mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;
mso-fareast-language:EN-US;mso-bidi-language:AR-IQ'>
m:val="p"/>M
style='mso-bidi-font-style:normal'>(
m:val="p"/>T ) is monoform module. Also, if
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EN-US;mso-bidi-language:AR-IQ'>
m:val="i"/>N( 𝒯) is a semi simple ring and
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mso-fareast-language:EN-US;mso-bidi-language:AR-IQ'>
m:val="p"/>M
style='mso-bidi-font-style:normal'>( 𝒯) is a
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EN-US;mso-bidi-language:AR-IQ'>
m:val="i"/>N( 𝒯)-module, so
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mso-fareast-font-family:Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:
"Times New Roman";mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;
mso-fareast-language:EN-US;mso-bidi-language:AR-IQ'>
m:val="p"/>M
style='mso-bidi-font-style:normal'>( 𝒯) is small and satisfies all conditions of monoform with Q-dedekind property. On the other hand, let
style='mso-bidi-font-style:normal'>
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mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>(M,
+, .) be an R-module.
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mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
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EN-US;mso-bidi-language:AR-IQ'>M
m:val="script"/>(
m:val="i"/>T
line-height:107%;font-family:"Cambria Math",serif;mso-fareast-font-family:
Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>)
m:val="script"/>
normal'> is a neutrosophic modules and generated by
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mso-fareast-font-family:Calibri;mso-fareast-theme-font:minor-latin;mso-bidi-font-family:
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m:val="p"/>M and
style='mso-bidi-font-style:normal'>
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mso-fareast-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:major-bidi;mso-ansi-language:EN-US;mso-fareast-language:
EN-US;mso-bidi-language:AR-IQ'>T . So,
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EN-US;mso-bidi-language:AR-IQ'>M
m:val="script"/>(
m:val="i"/>T)
is a weak neutrosophic. Finally, we presented more results, examples and properties about the topic with new results in neutrosophic algebra.
2025
316
321
3539
https://creativecommons.org/licenses/by/4.0/
10.54216/IJNS.250427
https://www.americaspg.com/journal/21/article/3539
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https://www.americaspg.com/storage/11738264684.pdf