1. CMB Online first
 Gao, Zenghui

Homological Properties Relative to Injectively Resolving Subcategories
Let $\mathcal{E}$ be an injectively resolving subcategory of
left $R$modules. A left $R$module $M$
(resp. right $R$module $N$) is called $\mathcal{E}$injective
(resp. $\mathcal{E}$flat)
if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$)
for any $G\in\mathcal{E}$.
Let $\mathcal{E}$ be a covering subcategory.
We prove that a left $R$module $M$ is $\mathcal{E}$injective
if and only if $M$ is a direct sum
of an injective left $R$module and a reduced $\mathcal{E}$injective
left $R$module.
Suppose $\mathcal{F}$ is a preenveloping subcategory of right
$R$modules such that
$\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$.
It is shown that a finitely presented right $R$module $M$ is
$\mathcal{E}$flat if and only if
$M$ is a cokernel of an $\mathcal{F}$preenvelope of a right
$R$module.
In addition, we introduce and investigate the
$\mathcal{E}$injective and $\mathcal{E}$flat dimensions of
modules and rings. We also introduce $\mathcal{E}$(semi)hereditary
rings and $\mathcal{E}$von Neumann regular rings and characterize
them in terms of $\mathcal{E}$injective and $\mathcal{E}$flat
modules.
Keywords:injectively resolving subcategory, \mathcal{E}injective module (dimension), \mathcal{E}flat module (dimension), cover, preenvelope, \mathcal{E}(semi)hereditary ring Categories:16E30, 16E10, 16E60 

2. CMB Online first
 Efrat, Ido; Matzri, Eliyahu

Vanishing of Massey products and Brauer groups
Let $p$ be a prime number and $F$ a field containing a root of
unity of order $p$.
We relate recent results on vanishing of triple Massey products
in the mod$p$ Galois cohomology of $F$,
due to Hopkins, Wickelgren, MinÃ¡Ä, and TÃ¢n, to classical
results in the theory of central simple algebras.
For global fields, we prove a stronger form of the vanishing
property.
Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields Categories:16K50, 11R34, 12G05, 12E30 

3. CMB 2015 (vol 58 pp. 233)
 Bergen, Jeffrey

Affine Actions of $U_q(sl(2))$ on Polynomial Rings
We classify the affine actions of $U_q(sl(2))$ on commutative
polynomial rings in $m \ge 1$ variables.
We show that, up to scalar multiplication, there are two possible
actions.
In addition, for each action, the subring of invariants is a
polynomial ring in either $m$ or $m1$ variables,
depending upon whether $q$ is or is not a root of $1$.
Keywords:skew derivation, quantum group, invariants Categories:16T20, 17B37, 20G42 

4. CMB 2015 (vol 58 pp. 263)
 De Filippis, Vincenzo; Mamouni, Abdellah; Oukhtite, Lahcen

Generalized Jordan Semiderivations in Prime Rings
Let $R$ be a ring, $g$ an endomorphism of $R$.
The additive mapping $d\colon R\rightarrow R$ is called Jordan semiderivation of $R$, associated with $g$, if
$$d(x^2)=d(x)x+g(x)d(x)=d(x)g(x)+xd(x)\quad \text{and}\quad d(g(x))=g(d(x))$$
for all $x\in R$.
The additive mapping $F\colon R\rightarrow R$ is called generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if
$$F(x^2)=F(x)x+g(x)d(x)=F(x)g(x)+xd(x)\quad \text{and}\quad F(g(x))=g(F(x))$$
for all $x\in R$.
In the present paper we prove that
if $R$ is a prime ring of characteristic different from $2$, $g$ an endomorphism of $R$, $d$ a Jordan semiderivation associated with $g$, $F$ a generalized Jordan semiderivation associated with $d$ and $g$,
then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R.$ Moreover, if $R$ is commutative then $F=d$.
Keywords:semiderivation, generalized semiderivation, Jordan semiderivation, prime ring Category:16W25 

5. CMB 2014 (vol 58 pp. 134)
6. CMB Online first
7. CMB 2014 (vol 57 pp. 814)
 Hou, Ruchen

On Global Dimensions of Tree Type Finite Dimensional Algebras
A formula is provided to
explicitly describe global dimensions of all kinds of tree type
finite dimensional $k$algebras for $k$ an algebraic closed field.
In particular, it is pointed out that if the underlying tree type
quiver has $n$ vertices, then the maximum of possible global
dimensions is $n1$.
Keywords:global dimension, tree type finite dimensional $k$algebra, quiver Categories:16D40, 16E10, , 16G20 

8. CMB 2014 (vol 57 pp. 609)
 NasrIsfahani, Alireza

Jacobson Radicals of Skew Polynomial Rings of Derivation Type
We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation
type.
Keywords:skew polynomial rings, Jacobson radical, derivation Categories:16S36, 16N20 

9. CMB 2014 (vol 57 pp. 511)
 Gonçalves, Daniel

Simplicity of Partial Skew Group Rings of Abelian Groups
Let $A$ be a ring with local units, $E$ a set of local units for $A$,
$G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of
$A$ that contain local units.
We show that $A\star_{\alpha} G$ is simple if and only if $A$ is
$G$simple and the center of the corner $e\delta_0 (A\star_{\alpha} G)
e \delta_0$ is a field for all $e\in E$. We apply the result to
characterize simplicity of partial skew group rings in two cases,
namely for partial skew group rings arising from partial actions by
clopen subsets of a compact set and partial actions on the set level.
Keywords:partial skew group rings, simple rings, partial actions, abelian groups Categories:16S35, 37B05 

10. CMB 2014 (vol 57 pp. 264)
 Dai, Li; Dong, Jingcheng

On Semisimple Hopf Algebras of Dimension $pq^n$
Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a
semisimple Hopf algebra of dimension $pq^n$ over an algebraically
closed field of characteristic $0$. This paper proves that $H$ must
possess one of the following structures: (1) $H$ is semisolvable;
(2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group
algebra of group $G$ of order $p$, and $R$ is a semisimple YetterDrinfeld
Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.
Keywords:semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld double Category:16W30 

11. CMB 2014 (vol 57 pp. 231)
 Bagherian, J.

On the Multiplicities of Characters in Table Algebras
In this paper we show that every module of a table algebra
can be considered as a faithful module of some quotient table
algebra.
Also we prove that every faithful module of a table algebra
determines a closed subset which is a cyclic group.
As a main result we give some information about multiplicities
of characters in table algebras.
Keywords:table algebra, faithful module, multiplicity of character Categories:20C99, 16G30 

12. CMB 2013 (vol 57 pp. 72)
 Grari, A.

Un Anneau Commutatif associÃ© Ã un design symÃ©trique
Dans les articles \cite{1}, \cite{2} et \cite{3}; l'auteur dÃ©veloppe une reprÃ©sentation
d'un plan projectif fini par un
anneau commutatif unitaire dont les propriÃ©tÃ©s algÃ©briques dÃ©pendent
de la structure gÃ©omÃ©trique du plan. Dans l'article \cite{4}; il Ã©tend cette reprÃ©sentation aux designs symÃ©triques. Cependant l'auteur de l'article \cite{7} fait remarquer que la multiplication dÃ©finie dans ce cas ne peut Ãªtre associative que si le design est un plan projectif.
Dans ce papier on mÃ¨nera
une Ã©tude de cette reprÃ©sentation dans le cas des designs
symÃ©triques. On y montrera comment on peut faire associer un
anneau commutatif unitaire Ã
tout design symÃ©trique , on y prÃ©cisera certaines de ses propriÃ©tÃ©s, en
particulier, celles qui relÃ¨vent de son invariance. On caractÃ©risera aussi les gÃ©omÃ©tries projectives finies de dimension supÃ©rieure moyennant cette reprÃ©sentation.
Keywords:projective planes, symmetric designs, commutative rings Categories:05B05, 16S99 

13. 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 

14. 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 

15. 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 

16. 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 

17. 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 

18. 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 

19. 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 

20. CMB 2011 (vol 55 pp. 271)
21. 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 

22. 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 

23. CMB 2011 (vol 55 pp. 208)
24. 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 

25. 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 
