Hongdi Huang, Pure Mathematics, University of Waterloo
"On *-clean group algebras"
A
ring
$R$
is
called
a
$*$-ring
(or
a
ring
with
involution
$*$)
if
there
exists
an
operation
$*$:
$R
\rightarrow
R$
such
that
$(x+y)^*=x^*+y^*,
\,\
(xy)^*=y^*x^*
\,\
$
and
$(x^*)^*=x$,
for
all
$x,
y\in
R$.
An
element
in
a
ring
$R$
is
called
$*$-clean
if
it
is
the
sum
of
a
unit
and
a
projection
($*$-invariant
idempotent).
A
$*$-ring
is
called
$*$-clean
if
each
of
its
elements
is
the
sum
of
a
unit
and
a
projection.
It is interesting to investigate the $*$-cleanness of $F_qG$, where $F_q$ is a finite field and $G$ is a finite cyclic group. Moreover, we will consider when the group algebras of the dihedral groups $D_{2n}$, and the generalized quaternion groups $Q_{2n}$ with standard involution $*$ over a ring $R$ are $*$-clean.