1. 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 

2. 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 

3. 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 

4. CMB 2013 (vol 57 pp. 270)
 Didas, Michael; Eschmeier, Jörg

Derivations on Toeplitz Algebras
Let $H^2(\Omega)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \subset
\mathbb{C}^n$,
and let $A \subset L^\infty(\partial \Omega)$ denote the subalgebra of all $L^\infty$functions $f$
with compact Hankel operator $H_f$. Given any closed subalgebra $B \subset A$ containing $C(\partial \Omega)$,
we describe the first Hochschild cohomology group of the
corresponding Toeplitz algebra $\mathcal(B) \subset B(H^2(\Omega))$.
In particular, we show that every derivation on $\mathcal{T}(A)$ is inner. These results are new even for $n=1$,
where it follows that every derivation on $\mathcal{T}(H^\infty+C)$ is inner, while there are noninner
derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over
the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$.
Keywords:derivations, Toeplitz algebras, strictly pseudoconvex domains Categories:47B47, 47B35, 47L80 

5. 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 

6. CMB 2012 (vol 56 pp. 534)
 Filali, M.; Monfared, M. Sangani

A Cohomological Property of $\pi$invariant Elements
Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$
be a continuous representation of $A$ on a separable Hilbert space $H$
with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of
$\pi$ with respect to an orthonormal basis and suppose that for each
$1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m}
\\pi_{ij}\_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these
conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$
left $\pi$invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all
$a\in A$. In this paper we prove a link between the existence
of left $\pi$invariant elements and the vanishing of certain
Hochschild cohomology groups of $A$. Our results extend an earlier
result by Lau on $F$algebras and recent results of KaniuthLauPym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a nonzero character on $A$.
Keywords:Banach algebras, $\pi$invariance, derivations, representations Categories:46H15, 46H25, 13N15 

7. CMB 2011 (vol 56 pp. 31)
 Ayuso, Fortuny P.

Derivations and Valuation Rings
A complete characterization of valuation rings closed for a
holomorphic derivation is given, following an idea of Seidenberg,
in dimension $2$.
Keywords:singular holomorphic foliation, derivation, valuation, valuation ring Categories:32S65, 13F30, 13A18 

8. CMB 2010 (vol 54 pp. 21)
 Bouali, S.; Echchad, M.

Generalized Dsymmetric Operators II
Let $H$ be a separable,
infinitedimensional, complex Hilbert space and let $A, B\in{\mathcal L
}(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear
operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal
L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AXXB$.
This note will initiate a study on the class of pairs $(A,B)$ such
that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal
R}(\delta_{A^{\ast}B^{\ast}})}$.
Keywords:generalized derivation, adjoint, Dsymmetric operator, normal operator Categories:47B47, 47B10, 47A30 

9. CMB 2009 (vol 52 pp. 535)
 Daigle, Daniel; Kaliman, Shulim

A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$
We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space Categories:14R10, 14R20, 14R25, 13N15 
