@article {Dolzan2005,
 author="David Dol\vzan",
 title={Complementation of the Jacobson group in a matrix ring},
 journal="Bull. Austral. Math. Soc.",
 fjournal={Bulletin of the Australian Mathematical Society},
 volume="72",
 year="2005",
 number="2",
 pages="317--324",
 issn="0004-9727",
 coden="ALNBAB",
 language="English",
 date="4th July, 2005",
 classmath="16N20, 16U60",
 publisher={AMPAI, Australian Mathematical Society},
 MRnumber="MR2183412",
 ZBLnumber="02246393",
 url="http://www.austms.org.au/Publ/Bulletin/V72P2/722-5197-Dolzan/index.shtml",
 acknowledgement={},
 abstract={ 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. }
}