Let Rbe a commutative local ring with the identity 1 and the unique maximal ideal  M be a free module of rank nover R, and s be in

Then, we factorize Minto a direct sum of mfree submodules such that each  is s-invariant modulo  for  and mis the number of the polynomials in the system of invariants of smodulo

Further it is shown that there exists a basis Xfor Mover Rfor which sis factorized into a product of elements in  in the form of

where each  is a cyclic permutation on X, each  is simple, i.e., fixes  elements in X. If 2 is a unit in R, then  can be replaced by a product of  symmetries in  As a result if 2 is a unit in R, then sis a product of nor less than nsimple elements.

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