aspg-24-976-1791482201 20261008175641 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Journal of Neutrosophic and Fuzzy Systems JNFS 2771-6430 2771-6449 2021 1 2 Katy D. Ahmad Islamic University Of Gaza, Palestine Mikail Bal Gaziantep University, Turkey Arwa A. Hajjari Cairo University, Egypt Rozina Ali Cairo University, Egypt This paper solves the imperfect duplets problem in refined neutrosophic rings, where it presents the necessary and sufficient conditions for a pair style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>( normal'> "Times New Roman";mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>x style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>, normal'> "Times New Roman";mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>y style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>) to be an imperfect duplet in any refined neutrosophic ring. Also, this work introduces a full description of the structure of imperfect duplets in numerical refined neutrosophic rings such as refined neutrosophic ring of integers normal'> "Times New Roman";mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>Z style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>( style='font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>I style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>1 style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>, style='font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>I style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>2 style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>) , refined neutrosophic ring of rationales style='mso-bidi-font-style:normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>Q normal'> "Times New Roman";mso-hansi-theme-font:major-bidi;mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>( style='font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>I style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>1 style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>, style='font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>I style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>2 style='mso-bidi-font-style:normal'> mso-hansi-font-family:"Times New Roman";mso-hansi-theme-font:major-bidi; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'>) , and refined neutrosophic ring or real numbers. 2021 71 79 976 https://creativecommons.org/licenses/by/4.0/ 10.54216/JNFS.010203 https://www.americaspg.com/journal/24/article/976 https://www.americaspg.com/storage/5987.pdf