It is well-known that an endomorphism of a finite-dimensional complex vector space is not completely determined by its eigenvalues or characteristic polynomial. In other words, not every endomorphism can be diagonalized. It is also well-known that one can express any endomorphism in Jordan Normal Form which is not far from diagonal. The characteristic polynomial can therefore be said to determine the endomorphism uniquely `up to choices of extensions' (in the sense of short exact sequences). More generally, if A is a commutative ring, Almkvist proved that an endomorphism of a finitely generated free A-module is determined uniquely up to extensions by its characteristic polynomial.
If A is a non-commutative ring then the familiar definitions of determinant and characteristic polynomial are unavailable. I shall define a version of the characteristic polynomial which makes sense for any ring A and which determines an endomorphism of a finitely generated free A-module uniquely up to extensions. Generalizing further, I'll give a characteristic "polynomial" for a system of two or more (non-commuting) endomorphisms of a free module.
The ideas behind this work come from topology: The relationship between an endomorphism and its characteristic polynomial can be considered analogous to the relationship between two classical approaches to knot theory.Desmond Sheiham