1. CMB Online first
 Lee, TsiuKwen

Adnilpotent elements of semiprime rings with involution
Let $R$ be an $n!$torsion free semiprime ring with
involution $*$ and with extended centroid $C$, where $n\gt 1$ is
a positive integer. We characterize $a\in K$, the Lie algebra
of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This
generalizes both Martindale and Miers' theorem
and the theorem of Brox et al.
To prove it we
first prove that if $a, b\in R$ satisfy
$(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on
$R$, where either $n$ is even or $b=0$, then
$\big(a\lambda\big)^{[\frac{n+1}{2}]}=0$
for some $\lambda\in C$.
Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$free, involution, skew (symmetric) element, adnilpotent element, Jordan element Categories:16N60, 16W10, 17B60 

2. CMB 2016 (vol 60 pp. 721)
 Eroǧlu, Münevver Pınar; Argaç, Nurcan

On Identities with Composition of Generalized Derivations
Let $R$ be a prime ring with extended
centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$
central closure of $R$ such that $dim_{C}(RC)
\gt 4$, $f(X_{1},\dots,X_{n})$
a multilinear polynomial over $C$ which is not centralvalued
on $R$ and $f(R)$ the set of all evaluations of the multilinear
polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that
$G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u
\in C$ for all $u\in f(R)$ then one of the following conditions
holds:
(I) there exists $a\in Q$ such that $a^2=0$ and
either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in
R$;
(II) there exists $a\in Q$ such that $0\neq a^2\in
C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all
$x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is centralvalued on
$R$;
(III) $char(R)=2$ and one of the following holds:
(i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$ and $a^{2}=b^{2}\in C$;
(ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$
is centralvalued on $R$;
(iii) there exist $a \in Q$ and an $X$outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and
$a^2+d(a)=0$;
(iv) there exist $a \in Q$ and an $X$outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$,
$a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is centralvalued
on $R$.
Moreover, we characterize the form of nonzero generalized derivations
$G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where
$\lambda \in C$.
Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotients Categories:16N60, 16N25 

3. CMB 2016 (vol 59 pp. 258)
 De Filippis, Vincenzo

Annihilators and Power Values of Generalized Skew Derivations on Lie Ideals
Let $R$ be a prime ring of characteristic different from
$2$, $Q_r$ be its right Martindale quotient ring and
$C$ be its extended centroid. Suppose that $F$ is
a generalized skew derivation of $R$, $L$ a noncentral Lie ideal
of $R$, $0 \neq a\in R$,
$m\geq 0$ and $n,s\geq 1$ fixed integers. If
\[
a\biggl(u^mF(u)u^n\biggr)^s=0
\]
for all $u\in L$, then either $R\subseteq M_2(C)$, the ring of
$2\times 2$ matrices over $C$, or $m=0$ and there exists $b\in
Q_r$ such that
$F(x)=bx$, for any $x\in R$, with $ab=0$.
Keywords:generalized skew derivation, prime ring Categories:16W25, 16N60 

4. 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 

5. 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 

6. 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 

7. CMB 2005 (vol 48 pp. 355)
8. CMB 2000 (vol 43 pp. 413)
9. 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 

10. 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 

11. 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 

12. CMB 1998 (vol 41 pp. 79)
13. 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 
