Let Rbe a commutative ring with 1, and M be a free module of a finite rank over R.

?is the endomorphism ring of Mover R, s isan element in ?and the matrix of sdiagonalizable. Our purpose is to investigate the relationship between the characteristic polynomial ?of sand the minimal polynomial ?of s. If Ris an integral domain, then we shall show that ?is uniquely determined as a monic polynomial dividing ?Also, the difference between the two sets of zeros of ?and ?respectively, is only the multiplicity of their roots. If Ris not an integral domain, then we shall construct ssuch that ?is not necessarily monic nor divides

minimal polynomial, characteristic polynomial, endomorphisms ring of modules, classical groups.