76. CMB 2000 (vol 43 pp. 413)
77. 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 

78. CMB 2000 (vol 43 pp. 100)
79. CMB 2000 (vol 43 pp. 79)
80. 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 

81. CMB 1999 (vol 42 pp. 298)
82. 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 

83. CMB 1999 (vol 42 pp. 371)
84. 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 

85. CMB 1998 (vol 41 pp. 481)
86. 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 

87. CMB 1998 (vol 41 pp. 359)
88. CMB 1998 (vol 41 pp. 261)
89. 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 

90. 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 

91. 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 

92. CMB 1998 (vol 41 pp. 79)
93. CMB 1997 (vol 40 pp. 221)
94. CMB 1997 (vol 40 pp. 198)
95. CMB 1997 (vol 40 pp. 47)
 Hartl, Manfred

A universal coefficient decomposition for subgroups induced by submodules of group algebras
Dimension subgroups and Lie dimension subgroups are known to satisfy a
`universal coefficient decomposition', {\it i.e.} their value with respect to
an arbitrary coefficient ring can be described in terms of their values with
respect to the `universal' coefficient rings given by the cyclic groups of
infinite and prime power order. Here this fact is generalized to much more
general types of induced subgroups, notably covering Fox subgroups and
relative dimension subgroups with respect to group algebra filtrations
induced by arbitrary $N$series, as well as certain common generalisations
of these which occur in the study of the former. This result relies on an
extension of the principal universal coefficient decomposition theorem on
polynomial ideals (due to Passi, Parmenter and Seghal), to all additive
subgroups of group rings. This is possible by using homological instead
of ring theoretical methods.
Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial ideals Categories:20C07, 16A27 

96. CMB 1997 (vol 40 pp. 103)
 Riley, David M.; Tasić, Vladimir

The transfer of a commutator law from a nilring to its adjoint group
For every field $F$ of characteristic $p\geq 0$,
we construct an example of a finite dimensional nilpotent
$F$algebra $R$ whose adjoint group $A(R)$ is not
centrebymetabelian, in spite of the fact that $R$ is Lie
centrebymetabelian
and satisfies the identities $x^{2p}=0$ when $p>2$ and
$x^8=0$ when $p=2$. The
existence of such algebras answers a question raised by
A.~E.~Zalesskii, and is in contrast to
positive results obtained by Krasilnikov, Sharma and Srivastava
for Lie metabelian rings
and by Smirnov for the class Lie centrebymetabelian nilalgebras
of exponent 4 over a field of characteristic 2 of cardinality at least 4.
Categories:16U60, 17B60 
