Jordan Decomposition

The Jordan canonical form for matrices over algebraically closed fields is standard fare in many linear algebra courses. The Jordan decomposition (into semisimple and nilpotent parts) for matrices over perfect fields is perhaps less well known, though very useful in many areas and closely related to the canonical form. This Jordan decomposition extends readily to elements of group algebras over perfect fields. During the past decade or so there has been activity in extending the decomposition to group rings (and matrices) over integral domains. In this article, we give a survey of this recent work (Arora et al., 1993 & 1998; Hales et al., 1990 & 1991) as well as some background on the classical results.