Results 221 to 230 of about 1,766,233 (264)
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Unit Groups of Commutative Group Rings
Communications in Algebra, 2006Let \(R\) be a unitary commutative ring and let \(G\) be an Abelian group. In the first part of the paper the authors assume that \(R\) is indecomposable and \(G\) is finite of order invertible in \(R\). They announce a result which describes the decomposition of the group ring \(RG\) into a direct sum of ring extensions of \(R\), in terms of the ...
Mollov, T., Nachev, N.
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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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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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GROUP ALGEBRAS WITH ENGEL UNIT GROUPS
Journal of the Australian Mathematical Society, 2016Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group.
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Free groups and involutions in the unit group of a group algebra
Archiv der Mathematik, 2005The authors study the existence of nonabelian free groups in the subgroup generated by the involutions of the unit group of the group algebra \(FG\) over a non-absolute field. If \(\text{char}(F)=p\neq 2\), \(G\) has no \(p\)-elements and the group of units \(U(FG)\) does not contain a nonabelian free subgroup, then the torsion part \(t(G)\) of \(G ...
GIAMBRUNO, Antonino, C. POLCINO MILIES
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1998
We saw in Chapter 5 that for each n, the set Un of units in ℤn forms a group under multiplication. Our aim in this chapter is to understand more about multiplication and division in ℤn by studying the structure of this group. An important result is that if n = pe, where p is an odd prime, then Un is cyclic; following a commonly-used strategy, we shall ...
Gareth A. Jones, J. Mary Jones
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We saw in Chapter 5 that for each n, the set Un of units in ℤn forms a group under multiplication. Our aim in this chapter is to understand more about multiplication and division in ℤn by studying the structure of this group. An important result is that if n = pe, where p is an odd prime, then Un is cyclic; following a commonly-used strategy, we shall ...
Gareth A. Jones, J. Mary Jones
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Alternating units as free factors in the group of units of integral group rings
Proceedings of the Edinburgh Mathematical Society, 2011AbstractLet G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in $\mathcal{U}(\mathbb{Z}G)$, the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large ...
Gonçalves, Jairo Z., Veloso, Paula M.
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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 Ideal Class Group and the Unit Group
1977Recall that the ideal class group of a number ring R consists of equivalence classes of nonzero ideals under the relation $$ I\sim J\quad iff\quad \alpha I = \beta J\quad for{\text{ }}some{\text{ }}non{\text{ }}zero{\text{ }}\alpha ,\beta \in R; $$ the group operation is multiplication defined in the obvious way, and the fact that this is ...
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Units in Group Algebras of Finite p-Groups
Journal of Mathematical Sciences, 2003zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Bovdi, Adalbert, Patay, Zoltán
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