Finite unitary ring with minimal non-nilpotent group of units
M Amiri, M Amini - Journal of Algebra and its Applications, 2021 - World Scientific
M Amiri, M Amini
Journal of Algebra and its Applications, 2021•World ScientificLet R be a finite unitary ring such that R= R 0 [R∗], where R 0 is the prime ring and R∗ is not
a nilpotent group. We show that if all proper subgroups of R∗ are nilpotent groups, then the
cardinality of R is a power of 2. In addition, if (R/Jac (R))∗ is not ap-group, then either R≅ M
2 (GF (2)) or R≅ M 2 (GF (2))⊕ A, where M 2 (GF (2)) is the ring of 2× 2 matrices over the
finite field GF (2) and A is a direct sum of copies of the finite field GF (2).
a nilpotent group. We show that if all proper subgroups of R∗ are nilpotent groups, then the
cardinality of R is a power of 2. In addition, if (R/Jac (R))∗ is not ap-group, then either R≅ M
2 (GF (2)) or R≅ M 2 (GF (2))⊕ A, where M 2 (GF (2)) is the ring of 2× 2 matrices over the
finite field GF (2) and A is a direct sum of copies of the finite field GF (2).
Let be a finite unitary ring such that , where is the prime ring and is not a nilpotent group. We show that if all proper subgroups of are nilpotent groups, then the cardinality of is a power of 2. In addition, if is not a -group, then either or , where is the ring of matrices over the finite field and is a direct sum of copies of the finite field .
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