What polynomial satisfies a given endomorphism? Original Research Article

The Authors study the ideal I(u) formed by the polynomial relations satisfied by an endomorphism u of a module M that has a finite free resolution. When M is torsion free, the ideal I(u) is characterized in terms of the characteristic polynomial χ(u,X) of u. This result gives a natural generalization of McCoyʹs theorem. The ideal I(u) is also related to the invariant factors of u. It is proved that in a large class of rings the ideal I(u) is principally generated by the last proper invariant factor of u. One obtains, for these cases, complete similarity with the classical case of fields.