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On the unit groups of rings with involution
Acta Mathematica Hungarica, 2022M. H. Bien, B. X. Hai, D. T. Hue
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Unit groups and Iwasawa lambda invariants of some multiquadratic number fields
, 2021M. M. Chems-Eddin, A. Azizi, A. Zekhnini
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Cyclotomic units and the unit group of an elementary abelian group ring
Archiv der Mathematik, 1985Let A be a finite abelian group, and let U(A) be the group of units of \({\mathbb{Z}}A\) modulo torsion. Consider the maps \[ \prod_{C}U(C)\to^{\alpha}U(A)\to^{\beta}\prod_{K}U(K) \] where C and K run over the sets of cyclic subgroups and factor-groups of A, respectively.
Hoechsmann, K., Sehgal, S. K., Weiss, A.
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Canadian Mathematical Bulletin, 1980
If R is a ring such that x, y ∈ R and xy = 0 imply yx = 0 and G≠ 1, an ordered group, then we show that ∑ αgg is a unit in RG if and only if there exists ∑ βhh in RG such that ∑ αgβg-1 = 1 and αgβh is nilpotent whenever gh≠l. We also show that if R is a ring with no nilpotent elements ≠ 0 and no idempotents ≠ 0, 1 then RG has only trivial units.
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If R is a ring such that x, y ∈ R and xy = 0 imply yx = 0 and G≠ 1, an ordered group, then we show that ∑ αgg is a unit in RG if and only if there exists ∑ βhh in RG such that ∑ αgβg-1 = 1 and αgβh is nilpotent whenever gh≠l. We also show that if R is a ring with no nilpotent elements ≠ 0 and no idempotents ≠ 0, 1 then RG has only trivial units.
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Group Identities on Units and Symmetric Units of Group Rings
2010Group Identities on Units of Group Rings.- Group Identities on Symmetric Units.- Lie Identities on Symmetric Elements.- Nilpotence of ...
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Group Rings With Hypercentral Unit Groups
Canadian Journal of Mathematics, 1991AbstractLet KG be the group ring of a group G over a field K and let U(KG) be its group of units. If K has characteristic p > 0 and G contains p-elements, then it is proved that U(KG) is hypercentral if and only if G is nilpotent and G′ is a finite p-group.
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2007
Summary: We give a complete characterization of the unit group \(\mathcal U(FS_3)\) of the group algebra \(FS_3\) of the symmetric group \(S_3\) of degree 3 over a finite field \(F\). Moreover, over the prime fields \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\), presentations of the unit groups of the group algebras \(\mathbb{Z}_2S_3\) and \(\mathbb{Z}_3S_3 ...
Sharma, R. K. +2 more
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Summary: We give a complete characterization of the unit group \(\mathcal U(FS_3)\) of the group algebra \(FS_3\) of the symmetric group \(S_3\) of degree 3 over a finite field \(F\). Moreover, over the prime fields \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\), presentations of the unit groups of the group algebras \(\mathbb{Z}_2S_3\) and \(\mathbb{Z}_3S_3 ...
Sharma, R. K. +2 more
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The Upper Central Series of the Unit Groups of Integral Group Rings: A Survey
, 2018Sugandha Maheshwary, I. Passi
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Group Algebras with Locally Nilpotent Unit Groups
, 2016M. Ramezan-Nassab
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