26. CMB 2013 (vol 57 pp. 506)
 Galindo, César

On Braided and Ribbon Unitary Fusion Categories
We prove that every braiding over a unitary fusion category is
unitary and every unitary braided fusion category admits a unique
unitary ribbon structure.
Keywords:fusion categories, braided categories, modular categories Categories:20F36, 16W30, 18D10 

27. CMB 2013 (vol 57 pp. 318)
 Huang, Zhaoyong

Duality of Preenvelopes and Pure Injective Modules
Let $R$ be an arbitrary ring and $()^+=\operatorname{Hom}_{\mathbb{Z}}(,
\mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers
and $\mathbb{Q}$ is the ring of rational numbers, and let
$\mathcal{C}$ be a subcategory of left $R$modules and $\mathcal{D}$
a subcategory of right $R$modules such that $X^+\in \mathcal{D}$
for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure
injective. Then a homomorphism $f: A\to C$ of left $R$modules with
$C\in \mathcal{C}$ is a $\mathcal{C}$(pre)envelope of $A$ provided
$f^+: C^+\to A^+$ is a $\mathcal{D}$(pre)cover of $A^+$. Some
applications of this result are given.
Keywords:(pre)envelopes, (pre)covers, duality, pure injective modules, character modules Categories:18G25, 16E30 

28. CMB 2013 (vol 57 pp. 159)
 Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal

Strongly $0$dimensional Modules
In a multiplication module, prime submodules have the property, if a prime
submodule contains a finite intersection of submodules then one of the
submodules is contained in the prime submodule. In this paper, we generalize
this property to infinite intersection of submodules and call such prime
submodules strongly prime submodule. A multiplication module in which every
prime submodule is strongly prime will be called strongly 0dimensional
module. It is also an extension of strongly 0dimensional rings. After
this we investigate properties of strongly 0dimensional modules and give
relations of von Neumann regular modules, Qmodules and strongly
0dimensional modules.
Keywords:strongly 0dimensional rings, Qmodule, Von Neumann regular module Categories:13C99, 16D10 

29. CMB 2012 (vol 57 pp. 51)
 Fošner, Ajda; Lee, TsiuKwen

Jordan $*$Derivations of FiniteDimensional Semiprime Algebras
In the paper, we characterize Jordan $*$derivations of a $2$torsion
free, finitedimensional semiprime algebra $R$ with involution $*$. To
be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan
$*$derivation. Then there exists a $*$algebra decomposition
$R=Uoplus V$ such that both $U$ and $V$ are invariant under
$delta$. Moreover, $*$ is the identity map of $U$ and $delta,_U$ is a
derivation, and the Jordan $*$derivation $delta,_V$ is inner.
We also prove the theorem: Let $R$ be a noncommutative, centrally
closed prime algebra with involution $*$, $operatorname{char},R
e 2$,
and let $delta$ be a nonzero Jordan $*$derivation of $R$. If $delta$ is
an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and
$delta$ is inner.
Keywords:semiprime algebra, involution, (inner) Jordan $*$derivation, elementary operator Categories:16W10, 16N60, 16W25 

30. CMB 2012 (vol 56 pp. 584)
 Liau, PaoKuei; Liu, ChengKai

On Automorphisms and Commutativity in Semiprime Rings
Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xyyx$ the commutator of
$x$ and $y$. If $\sigma$ is a nonidentity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.
Keywords:automorphism, generalized polynomial identity (GPI) Categories:16N60, 16W20, 16R50 

31. CMB 2011 (vol 56 pp. 564)
 Herzog, Ivo

Ziegler's Indecomposability Criterion
Ziegler's Indecomposability Criterion is used to prove that a totally
transcendental, i.e., $\Sigma$pure injective, indecomposable left
module over a left noetherian ring is a directed union of finitely
generated indecomposable modules. The same criterion is also used to
give a sufficient condition for a pure injective indecomposable module
${_R}U$ to have an indecomposable local dual $U_R^{\sharp}.$
Keywords:pure injective indecomposable module, local dual, generic module, amalgamation Categories:16G10, 03C60 

32. CMB 2011 (vol 56 pp. 344)
 Goodaire, Edgar G.; Milies, César Polcino

Involutions and Anticommutativity in Group Rings
Let $g\mapsto g^*$ denote an involution on a
group $G$. For any (commutative, associative) ring
$R$ (with $1$), $*$ extends linearly to an involution
of the group ring $RG$. An element $\alpha\in RG$
is symmetric if $\alpha^*=\alpha$ and
skewsymmetric if $\alpha^*=\alpha$.
The skewsymmetric elements are closed under
the Lie bracket, $[\alpha,\beta]=\alpha\beta\beta\alpha$.
In this paper, we investigate when this set is also closed
under the ring product in $RG$.
The symmetric elements are closed under the Jordan
product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$.
Here, we determine when this product is trivial.
These two problems
are analogues of problems about the skewsymmetric and
symmetric elements in group rings that have received a
lot of attention.
Categories:16W10, 16S34 

33. CMB 2011 (vol 55 pp. 271)
34. CMB 2011 (vol 55 pp. 579)
 Ndogmo, J. C.

Casimir Operators and Nilpotent Radicals
It is shown that a Lie algebra having a nilpotent radical has a
fundamental set of invariants consisting of Casimir operators. A
different proof is given in the well known special case of an
abelian radical. A result relating the number of invariants to the
dimension of the Cartan subalgebra is also established.
Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants Categories:16W25, 17B45, 16S30 

35. CMB 2011 (vol 55 pp. 260)
 Delvaux, L.; Van Daele, A.; Wang, Shuanhong

A Note on the Antipode for Algebraic Quantum Groups
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a coFrobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf
algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a
finitedimensional Hopf algebra, but also that of any
Hopf algebra with integrals (coFrobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from wellknown formulas obtained earlier in the
theory of regular multiplier Hopf algebras with integrals.
We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode Categories:16W30, 46L65 

36. CMB 2011 (vol 55 pp. 208)
37. CMB 2010 (vol 54 pp. 237)
 Creedon, Leo; Gildea, Joe

The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$
Let $RG$ denote the group ring of the group $G$ over
the ring $R$. Using an isomorphism between $RG$ and a
certain ring of $n \times n$ matrices in conjunction with other
techniques, the structure of the unit group of the group algebra
of the dihedral group of order $8$ over any
finite field of chracteristic $2$ is determined in
terms of split extensions of cyclic groups.
Categories:16U60, 16S34, 20C05, 15A33 

38. CMB 2010 (vol 53 pp. 587)
 Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq

Hulls of Ring Extensions
We investigate the behavior of the quasiBaer and the
right FIextending right ring hulls under various ring extensions
including group ring extensions, full and triangular matrix ring
extensions, and infinite matrix ring extensions. As a consequence,
we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are
Morita equivalent, then so are the quasiBaer right ring hulls
$\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of
$R$ and $S$, respectively. As an application, we prove that if
unital $C^*$algebras $A$ and $B$ are Morita equivalent as rings,
then the bounded central closure of $A$ and that of $B$ are
strongly Morita equivalent as $C^*$algebras. Our results show
that the quasiBaer property is always preserved by infinite
matrix rings, unlike the Baer property. Moreover, we give an
affirmative answer to an open question of Goel and Jain for the
commutative group ring $A[G]$ of a torsionfree Abelian group $G$
over a commutative semiprime quasicontinuous ring $A$. Examples
that illustrate and delimit the results of this paper are provided.
Keywords:(FI)extending, Morita equivalent, ring of quotients, essential overring, (quasi)Baer ring, ring hull, u.p.monoid, $C^*$algebra Categories:16N60, 16D90, 16S99, 16S50, 46L05 

39. CMB 2010 (vol 53 pp. 223)
 Chuang, ChenLian; Lee, TsiuKwen

Density of Polynomial Maps
Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil onesided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords:density, polynomial, endomorphism ring, PI Categories:16D60, 16S50 

40. CMB 2009 (vol 53 pp. 321)
 Lee, TsiuKwen; Zhou, Yiqiang

A Theorem on UnitRegular Rings
Let $R$ be a unitregular ring and let $\sigma $ be an endomorphism of
$R$ such that $\sigma (e)=e$ for all $e^2=e\in R$ and let $n\ge 0$. It
is proved that every element of $R[x \mathinner;\sigma]/(x^{n+1})$ is
equivalent to an element of the form $e_0+e_1x+\dots +e_nx^n$, where
the $e_i$ are orthogonal idempotents of $R$. As an application, it is
proved that $R[x \mathinner; \sigma ]/(x^{n+1})$ is left morphic for each
$n\ge 0$.
Keywords:morphic rings, unitregular rings, skew polynomial rings Categories:16E50, 16U99, 16S70, 16S35 

41. CMB 2009 (vol 53 pp. 230)
42. CMB 2009 (vol 52 pp. 564)
 Jin, Hai Lan; Doh, Jaekyung; Park, Jae Keol

Group Actions on QuasiBaer Rings
A ring $R$ is called {\it quasiBaer} if the right
annihilator of every right ideal of $R$ is generated by an
idempotent as a right ideal. We investigate the quasiBaer
property of skew group rings and fixed rings under a finite group
action on a semiprime ring and their applications to
$C^*$algebras.
Various examples to illustrate and
delimit our results are provided.
Keywords:(quasi) Baer ring, fixed ring, skew group ring, $C^*$algebra, local multiplier algebra Categories:16S35, 16W22, 16S90, 16W20, 16U70 

43. 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 

44. 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 

45. 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 

46. CMB 2008 (vol 51 pp. 424)
47. 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 

48. 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 

49. 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 

50. 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 
