51. CMB 2009 (vol 40 pp. 221)
52. CMB 2009 (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 

53. CMB 2009 (vol 52 pp. 267)
 Ko\c{s}an, Muhammet Tamer

Extensions of Rings Having McCoy Condition
Let $R$ be an associative ring with unity.
Then $R$ is said to be a {\it right McCoy ring} when the equation
$f(x)g(x)=0$ (over $R[x]$), where $0\neq f(x),g(x) \in R[x]$,
implies that there exists a nonzero element $c\in R$ such that
$f(x)c=0$. In this paper, we characterize some basic ring
extensions of right McCoy rings and we prove that if $R$ is a
right McCoy ring, then $R[x]/(x^n)$ is
a right McCoy ring for any positive integer $n\geq 2$ .
Keywords:right McCoy ring, Armendariz ring, reduced ring, reversible ring, semicommutative ring Categories:16D10, 16D80, 16R50 

54. CMB 2009 (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 

55. CMB 2009 (vol 52 pp. 145)
 Wang, Z.; Chen, J. L.

$2$Clean Rings
A ring $R$ is said to be $n$clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$module of rank at least 2 is $2$clean and that the ring $B(R)$
of all $\omega\times \omega$ row and columnfinite matrices over
any ring $R$ is $2$clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.
Keywords:$2$clean rings, $2$good rings, free modules, row and columnfinite matrix rings, group rings Categories:16D70, 16D40, 16S50 

56. CMB 2009 (vol 52 pp. 39)
 Cimpri\v{c}, Jakob

A Representation Theorem for Archimedean Quadratic Modules on $*$Rings
We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
GelfandNaimark representation theorem for commutative $C^\ast$algebras.
A noncommutative version of GelfandNaimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Keywords:Ordered rings with involution, $C^\ast$algebras and their representations, noncommutative convexity theory, real algebraic geometry Categories:16W80, 46L05, 46L89, 14P99 

57. CMB 2008 (vol 51 pp. 424)
58. CMB 2008 (vol 51 pp. 460)
 Smoktunowicz, Agata

On Primitive Ideals in Graded Rings
Let $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown
that primitive ideals in $R$ are homogeneous. Let
$A=\bigoplus_{i=1}^{\infty}A_{i}$ be a graded nonPI justinfinite
dimensional algebra and let $I$ be a prime ideal in $A$. It is shown
that either $I=\{0\}$ or $I=A$. Moreover, $A$ is either primitive or
Jacobson radical.
Categories:16D60, 16W50 

59. CMB 2008 (vol 51 pp. 261)
 Neeb, KarlHermann

On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups
An $n$dimensional quantum torus is a twisted group algebra of the
group $\Z^n$. It is called rational if all invertible commutators are roots
of unity. In the present note we describe a normal form for rational
$n$dimensional quantum
tori over any field. Moreover, we show that for
$n = 2$ the natural exact sequence
describing the automorphism group of the quantum torus splits over any
field.
Keywords:quantum torus, normal form, automorphisms of quantum tori Category:16S35 

60. CMB 2008 (vol 51 pp. 291)
 Spinelli, Ernesto

Group Algebras with Minimal Strong Lie Derived Length
Let $KG$ be a noncommutative strongly Lie solvable group algebra of a
group $G$ over a field $K$ of positive characteristic $p$. In this
note we state necessary and sufficient conditions so that the
strong Lie derived length of $KG$ assumes its minimal value, namely
$\lceil \log_{2}(p+1)\rceil $.
Keywords:group algebras, strong Lie derived length Categories:16S34, 17B30 

61. CMB 2008 (vol 51 pp. 81)
 Kassel, Christian

Homotopy Formulas for Cyclic Groups Acting on Rings
The positive cohomology groups of a finite group acting on a ring
vanish when the ring has a norm one element. In this note we give
explicit homotopies on the level of cochains when the group is cyclic,
which allows us to express any cocycle of a cyclic group
as the coboundary of an explicit cochain.
The formulas in this note are closely related to the effective problems considered in previous joint work
with Eli Aljadeff.
Keywords:group cohomology, norm map, cyclic group, homotopy Categories:20J06, 20K01, 16W22, 18G35 

62. CMB 2007 (vol 50 pp. 105)
 Klep, Igor

On Valuations, Places and Graded Rings Associated to $*$Orderings
We study natural $*$valuations, $*$places and graded $*$rings
associated with $*$ordered rings.
We prove that the natural $*$valuation is always quasiOre and is
even quasicommutative (\emph{i.e.,} the corresponding graded $*$ring is
commutative), provided the ring contains an imaginary unit.
Furthermore, it is proved that the graded $*$ring is isomorphic
to a twisted semigroup algebra. Our results are applied to answer a question
of Cimpri\v c regarding $*$orderability of quantum
groups.
Keywords:$*$orderings, valuations, rings with involution Categories:14P10, 16S30, 16W10 

63. CMB 2006 (vol 49 pp. 347)
 Ecker, Jürgen

Affine Completeness of Generalised Dihedral Groups
In this paper we study affine completeness of generalised dihedral
groups. We give a formula for the number of unary compatible
functions on these groups, and we characterise for every $k \in~\N$
the $k$affine complete generalised dihedral groups. We find that
the direct product of a $1$affine complete group with itself need not
be $1$affine complete. Finally, we give an example of a nonabelian
solvable affine complete group. For nilpotent groups we find a
strong necessary condition for $2$affine completeness.
Categories:08A40, 16Y30, 20F05 

64. CMB 2006 (vol 49 pp. 265)
 Nicholson, W. K.; Zhou, Y.

Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial
If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in
C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$clean if every element is
the sum of a unit and a root of $g(x)$. If $V$ is a vector space of
countable dimension over a division ring $D,$ they showed that
$\end {}_{D}V$ is
$g(x)$clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=xx^{2}$
this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan.
In this paper we remove the countable condition, and in fact prove that
$\Mend {}_{R}M$ is $g(x)$clean for any semisimple module $M$ over an arbitrary
ring $R$ provided that $g(x)\in (xa)(xb)C[x]$ where $a,b\in C$ and both $b$
and $ba$ are units in $R$.
Keywords:Clean rings, linear transformations, endomorphism rings Categories:16S50, 16E50 

65. CMB 2005 (vol 48 pp. 587)
 Lopes, Samuel A.

Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$
Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping
algebra $U_{q}(\SL)$. Using results of AlevDumas and Caldero related
to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free
over its center. This is reminiscent of Kostant's separation of
variables for the enveloping algebra $U(\g)$ of a complex semisimple
Lie algebra $\g$, and also of an analogous result of JosephLetzter
for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to
its representation theory is the fact that $\U{+}$ is free over a
larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$
to $\U{+}$ provides infinitedimensional modules with good properties,
including a grading that is inherited by submodules.
Categories:17B37, 16W35, 17B10, 16D60 

66. CMB 2005 (vol 48 pp. 445)
 Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred

On the Garsia Lie Idempotent
The orthogonal projection of the free associative algebra onto the
free Lie algebra is afforded by an idempotent in the rational group
algebra of the symmetric group $S_n$, in each homogenous degree
$n$. We give various characterizations of this Lie idempotent and show
that it is uniquely determined by a certain unit in the group algebra
of $S_{n1}$. The inverse of this unit, or, equivalently, the Gram
matrix of the orthogonal projection, is described explicitly. We also
show that the Garsia Lie idempotent is not constant on descent classes
(in fact, not even on coplactic classes) in $S_n$.
Categories:17B01, 05A99, 16S30, 17B60 

67. CMB 2005 (vol 48 pp. 355)
68. CMB 2005 (vol 48 pp. 275)
 Smith, Patrick F.

Krull Dimension of Injective Modules Over Commutative Noetherian Rings
Let $R$ be a commutative Noetherian
integral domain with field of fractions $Q$. Generalizing a
fortyyearold theorem of E. Matlis, we prove that the $R$module
$Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal
and onedimensional. Moreover, if $X$ is an injective module over
a commutative Noetherian ring such that $X$ has Krull dimension,
then the Krull dimension of $X$ is at most $1$.
Categories:13E05, 16D50, 16P60 

69. CMB 2005 (vol 48 pp. 317)
70. CMB 2005 (vol 48 pp. 80)
 Herman, Allen; Li, Yuanlin; Parmenter, M. M.

Trivial Units for Group Rings with $G$adapted Coefficient Rings
For each finite group $G$ for which the integral group ring
$\mathbb{Z}G$ has only trivial units, we give ringtheoretic
conditions for a commutative ring $R$ under which the group ring
$RG$ has nontrivial units. Several examples of rings satisfying
the conditions and rings not satisfying the conditions are given.
In addition, we extend a wellknown result for fields by showing
that if $R$ is a ring of finite characteristic and $RG$ has only
trivial units, then $G$ has order at most 3.
Categories:16S34, 16U60, 20C05 

71. CMB 2004 (vol 47 pp. 445)
 Pirkovskii, A. Yu.

Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators
For a locally compact group $G$, the convolution product on
the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang
\cite{Neuf_PhD}. We study homological properties of the convolution algebra
$\nN(L^p(G))$ and relate them to some properties of the group $G$,
such as compactness, finiteness, discreteness, and amenability.
Categories:46M10, 46H25, 43A20, 16E65 

72. CMB 2004 (vol 47 pp. 343)
 Drensky, Vesselin; Hammoudi, Lakhdar

Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras
We construct new examples of nonnil algebras with any number of
generators, which are direct sums of two
locally nilpotent subalgebras. Like all previously known examples, our examples
are contracted semigroup algebras and the underlying semigroups are unions
of locally nilpotent subsemigroups.
In our constructions we make more
transparent
than in the past the close relationship between the considered problem
and combinatorics of words.
Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words Categories:16N40, 16S15, 20M05, 20M25, 68R15 

73. CMB 2003 (vol 46 pp. 14)
74. CMB 2002 (vol 45 pp. 451)
 Allison, Bruce; Smirnov, Oleg

Coordinatization Theorems For Graded Algebras
In this paper we study simple associative algebras with finite
$\mathbb{Z}$gradings. This is done using a simple algebra $F_g$
that has been constructed in Morita theory from a bilinear form
$g\colon U\times V\to A$ over a simple algebra $A$. We show that
finite $\mathbb{Z}$gradings on $F_g$ are in one to one
correspondence with certain decompositions of the pair $(U,V)$. We
also show that any simple algebra $R$ with finite
$\mathbb{Z}$grading is graded isomorphic to $F_g$ for some
bilinear from $g\colon U\times V \to A$, where the grading on $F_g$
is determined by a decomposition of $(U,V)$ and the coordinate
algebra $A$ is chosen as a simple ideal of the zero component $R_0$
of $R$. In order to prove these results we first prove similar
results for simple algebras with Peirce gradings.
Category:16W50 

75. 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 
