Consider the action of $GL_n(R)$ on $M_{n \times n}(R)$ by conjugation, where $R$ is a ring (or field)? How can we classify the orbits? To what extent does the characteristic polynomial and the trace classify the orbits?
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2$\begingroup$ Many papers on linear algebra or algebraic K-theory study problems about linear algebra over rings; there is even a 1984 book Linear algebra over commutative rings by B.R. McDonald. But the problems here typically are much harder than those over fields even when the ring is quite special. $\endgroup$– Jim HumphreysCommented Mar 9, 2011 at 16:10
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For a field, this is given by the rational canonical form (see Section 7.2 of Hoffman and Kunze's Linear Algebra, for example). Even in this case, the trace and characteristic polynomial are quite weak as invariants. What you need are the invariant factors. For general rings, this is very hard and often a wild classifcation problem. For rings like $\mathbf Z/p^k\mathbf Z$ look at this paper.
answered Mar 9, 2011 at 7:37