76. CMB 2002 (vol 45 pp. 711)
 Yoshii, Yoji

Classification of Quantum Tori with Involution
Quantum tori with graded involution appear as coordinate algebras of
extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$.
We classify them in the category of algebras with involution. From
this, we obtain precise information on the root systems of extended
affine Lie algebras of type $\rmC$.
Category:16W50 

77. CMB 2002 (vol 45 pp. 499)
 Bahturin, Yu. A.; Zaicev, M. V.

Group Gradings on Matrix Algebras
Let $\Phi$ be an algebraically closed field of characteristic zero,
$G$ a finite, not necessarily abelian, group. Given a $G$grading on
the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the
tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p
(\Phi)$ being a graded division algebra, while the grading of $C\cong
M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now
the grading of $A$ is recovered from those of $A$ and $B$ using a
canonical ``induction'' procedure.
Category:16W50 

78. CMB 2002 (vol 45 pp. 448)
79. CMB 2002 (vol 45 pp. 388)
 Gille, Philippe

AlgÃ¨bres simples centrales de degrÃ© 5 et $E_8$
As a consequence of a theorem of RostSpringer, we establish that the
cyclicity problem for central simple algebra of degree~5 on fields
containg a fifth root of unity is equivalent to the study of
anisotropic elements of order 5 in the split group of type~$E_8$.
Keywords:algÃ¨bres simples centrales, cohomologie galoisienne Categories:16S35, 12G05, 20G15 

80. CMB 2002 (vol 45 pp. 11)
81. CMB 2001 (vol 44 pp. 27)
82. CMB 2000 (vol 43 pp. 413)
83. CMB 2000 (vol 43 pp. 3)
 Adin, Ron; Blanc, David

Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and
free Lie algebras in the category of nonassociative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaflabeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, nonassociative algebras, Jacobi identity, leaflabeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 

84. CMB 2000 (vol 43 pp. 60)
 Farkas, Daniel R.; Linnell, Peter A.

Trivial Units in Group Rings
Let $G$ be an arbitrary group and let $U$ be a subgroup of the
normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$
as a subgroup of finite index, then $U = G$. This result can be used
to give an alternative proof of a recent result of Marciniak and
Sehgal on units in the integral group ring of a crystallographic group.
Keywords:units, trace, finite conjugate subgroup Categories:16S34, 16U60 

85. CMB 2000 (vol 43 pp. 79)
86. CMB 2000 (vol 43 pp. 100)
87. CMB 1999 (vol 42 pp. 401)
 Swain, Gordon A.; Blau, Philip S.

Lie Derivations in Prime Rings With Involution
Let $R$ be a nonGPI prime ring with involution and characteristic
$\neq 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote
the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$
into itself. Then $\delta=\rho+\epsilon$ where $\epsilon$ is an
additive map into the skew elements of the extended centroid of $R$
which is zero on $[K,K]$, and $\rho$ can be extended to an ordinary
derivation of $\langle K\rangle$ into $RC$, the central closure.
Categories:16W10, 16N60, 16W25 

88. CMB 1999 (vol 42 pp. 298)
89. CMB 1999 (vol 42 pp. 371)
90. CMB 1999 (vol 42 pp. 174)
 Ferrero, Miguel; Sant'Ana, Alveri

Rings With Comparability
The class of rings studied in this paper properly contains the
class of right distributive rings which have at least one
completely prime ideal in the Jacobson radical. Amongst other
results we study prime and semiprime ideals, right noetherian rings
with comparability and prove a structure theorem for rings with
comparability. Several examples are also given.
Categories:16U99, 16P40, 16D14, 16N60 

91. CMB 1998 (vol 41 pp. 452)
 Brešar, Matej; Martindale, W. S.; Miers, C. Robert

Dependent automorphisms in prime rings
For each $n\geq 4$ we construct a class of examples of a minimal
$C$dependent set of $n$ automorphisms of a prime ring $R$, where $C$
is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that
the preceding examples are completely general, whereas for $n=6$ an
example is given which fails to enjoy any of the nice properties of
the above example.
Categories:16N60, 16W20 

92. CMB 1998 (vol 41 pp. 481)
93. CMB 1998 (vol 41 pp. 261)
94. CMB 1998 (vol 41 pp. 359)
95. CMB 1998 (vol 41 pp. 79)
96. CMB 1998 (vol 41 pp. 118)
 Valenti, Angela

On permanental identities of symmetric and skewsymmetric matrices in characteristic \lowercase{$p$}
Let $M_n(F)$ be the algebra of $n \times n$
matrices over a field $F$ of characteristic $p>2$ and let $\ast$ be an
involution on $M_n(F)$. If $s_1, \ldots, s_r$ are symmetric
variables we determine the smallest $r$ such that the polynomial
$$
P_{r}(s_1, \ldots, s_{r}) = \sum_{\sigma \in {\cal
S}_r}s_{\sigma(1)}\cdots s_{\sigma(r)}
$$
is a $\ast$polynomial identity of $M_n(F)$ under either the
symplectic or the transpose involution. We also prove an analogous
result for the polynomial
$$
C_r(k_1, \ldots, k_r, k'_1, \ldots, k'_r) = \sum_
{\sigma, \tau \in {\cal S}_r}k_{\sigma(1)}k'_{\tau(1)}\cdots
k_{\sigma(r)}k'_{\tau(r)}
$$
where $k_1, \ldots, k_r, k'_1, \ldots, k'_r$ are skew
variables under the transpose involution.
Category:16R50 

97. CMB 1998 (vol 41 pp. 109)
 Tahara, KenIchi; Vermani, L. R.; Razdan, Atul

On generalized third dimension subgroups
Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M.~Hartl
identified the subgroup $G \cap(1+\triangle^3(G)+\triangle(G)\triangle(H))$
of $G$. In this note we give an independent proof of the result of Hartl,
and we identify two subgroups
$G\cap(1+\triangle(H)\triangle(G)\triangle(H)+\triangle([H,G])\triangle(H))$,
$G\cap(1+\triangle^2(G)\triangle(H)+\triangle(K)\triangle(H))$ of $G$ for
some subgroup $K$ of $G$ containing $[H,G]$.
Categories:20C07, 16S34 

98. CMB 1998 (vol 41 pp. 81)
 Lanski, Charles

The cardinality of the center of a $\PI$ ring
The main result shows that if $R$ is a semiprime ring satisfying
a polynomial identity, and if $Z(R)$ is the center of $R$, then
$\card R \leq 2^{\card Z(R)}$. Examples show that this bound can
be achieved, and that the inequality fails to hold for rings which
are not semiprime.
Categories:16R20, 16N60, 16R99, 16U50 
