We study a representation of an arbitrary endomorphism son a free module Mof a finite rank nover a valuation domain R, that is, factorize Minto a direct sum of some free submodules determined by s, give a sufficient condition for sto have the rational canonical form, find a basis for Mover Rrelative to which the matrix of sis a sum of a lower triangular matrix and a super diagonal matrix, and show that for some simple ring extension of Rthe natural prolongation of shas the rational canonical form.

rational canonical form, system of invariants, Jordan normal form, minimal polynomial, characteristic polynomial, endomorphism ring of a module, classical groups.