aspg-21-1056-1791475391 20261008160311 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2022 18 3 Basheer Abd Al Rida Sadi Al-Imam Al-Kadhum College For Islamic Science University, Iraq This paper is dedicated to study the group of units problem in some 2-cyclic refined neutrosophic rings, where it presents a full classification of the group of units in 2-cyclic refined neutrosophic ring of integers style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>Z style='mso-bidi-font-style:normal'> 115%;font-family:"Cambria Math","serif";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>(I) , 2-cyclic refined neutrosophic ring of integers modulo 2 and 3 style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size: 10.0pt;font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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'> line-height:115%;font-family:"Cambria Math","serif";mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>Z style='mso-bidi-font-style:normal'> line-height:115%;font-family:"Cambria Math","serif";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>2 style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>I , normal'> mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:major-bidi; mso-bidi-language:AR-SY'> style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>( style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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'> line-height:115%;font-family:"Cambria Math","serif";mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>Z style='mso-bidi-font-style:normal'> line-height:115%;font-family:"Cambria Math","serif";mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>3 style='mso-bidi-font-style:normal'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>) style='mso-bidi-font-style:normal'> 115%;font-family:"Cambria Math","serif";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>(I) and 2-cyclic refined neutrosophic ring of real numbers style='font-size:10.0pt;mso-ansi-font-size:10.0pt;mso-bidi-font-size:10.0pt; font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>R style='mso-bidi-font-style:normal'> 115%;font-family:"Cambria Math","serif";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'> 115%;font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;mso-bidi-language:AR-SY'>(I) , as direct products of the cyclic groups. On the other hand, this work provides a necessary and sufficient condition for the invertibility of any square 2-cyclic refined neutrosophic real matrix. Also, we will illustrate some examples to clarify the validity of our work. 2022 48 58 1056 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.180304 https://www.americaspg.com/journal/21/article/1056 https://www.americaspg.com/storage/2600.pdf