Ring in Which Every Element is Sum of Two 5-Potent Elements
DOI:
https://doi.org/10.26713/cma.v15i1.2333Keywords:
5-Potents, Chinese Remainder Theorem, Jacobson radicalAbstract
Every element of a ring \(R\) is a sum of two commuting 5-potents if and only if \(R\cong R_1\times R_2\times R_3\times R_4\), where \(R_1/J(R_1)\) is Boolean and \(U(R_1)\) is a group of exponent \(4\), \(R_2\) is a subdirect product of \(Z_3\)'s, \(R_3\) is a subdirect product of \(Z_5\)'s and \(R_4\) is a subdirect product of \(Z_{13}\)'s. Also, if in a ring \(R\) every element is a sum of two 5-potents and a nilpotent that commute with one another then \(R\cong R_1\times R_2\times R_3\times R_4\) where \(R_1/J(R_1)\) is Boolean and \(J(R_1)\) is nil, \(R_2\cong R_a\times R_b\times R_c\) where \(R_a=0\), \(R_c=0\) and \(R_b/J(R_b)\) is a subdirect product of rings isomorphic to \(Z_3\), \(M_2(Z_3)\) or \(F_9\) with \(J(R_b)\) is nil, \(R_3/J(R_3)\) is a subdirect product of \(Z_5\)'s and \(J(R_3)\) is nil, \(R_4/J(R_4)\) is a subdirect product of \(Z_{13}\)'s and \(J(R_4)\) is nil.
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