Bulletin of the Australian Mathematical
Society
Bull. Austral. Math. Soc.
0004-9727
ALNBAB
2005
72
2
10.wxyz/CV72P2
http://www.austms.org.au/Publ/Bulletin/V72P2/
Complementation of the Jacobson group in a matrix
ring
David
Dolžan
14 February 2006
2006
2
14
317
324
722-5197-Dolzan-2005
10.wxyz/C2005V72P2p317
http://www.austms.org.au/Publ/Bulletin/V72P2/722-5197-Dolzan/
The Jacobson group of a ring $R$ (denoted by $\cJ
=\cJ (R)$) is the normal subgroup of the group of units of
$R$ (denoted by $G(R)$) obtained by adding 1 to the Jacobson
radical of $R$ $\bigl (J(R)\bigr )$. Coleman and Easdown in
2000 showed that the Jacobson group is complemented in the
group of units of any finite commutative ring and also in the
group of units of a $n\times n$ matrix ring over integers
modulo $p^s$, when $n=2$ and $p=2,3$, but it is not
complemented when $p\ge 5$. In 2004 Wilcox showed that the
answer is positive also for $n=3$ and $p=2$, and negative in
all the remaining cases. In this paper we offer a different
proof for Wilcox's results and also generalise the results to
a matrix ring over an arbitrary finite commutative ring. We
show this by studying the generators and relations that
define a matrix ring over a field. We then proceed to examine
the complementation of the Jacobson group in the matrix rings
over graded rings and prove that complementation depends only
on the 0-th grade.
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