Corollary 5

Let K be a commutative field, V a finite-dimensional vector space over K, and $u$ and $v$ endomorphisms of V. The following properties are equivalent:

(i) There exists a polynomial P in K[X] such that $v= P(u)$.

(ii) The endomorphism $v$ commutes with every endomorphism of V that commutes with $u$.

Take A to be the ring K[X] and M to be the K[X]-module deduced from V and $u$ (VII, §5, No. 1, p. 28). Assertion (i) means that $v\in K[X]_M$ and (ii) that $v\in K[X]''_M$. Corollary 5 is therefore a particular case of Corollary 4.