Length problems for automorphisms of modules over local rings

Let R be a local ring and M a free module of a finite rank over R. An element τ AutRM is said to be simple if τ ≠ 1 fixes a hyperplane of M. We shall show that for any σ AutRM there exist a basis X for M and ρ AutRM such that ρ acts as a permutation on X and ρ−1σ is a product of m or less than m simple elements in AutRM, where m is the order of the invariant factors of σ modulo the maximal ideal of R. Also we shall investigate the problem treated by E.W. Ellers and H. Ishibashi [Factorizations of transformations over a valuation ring, Linear Algebra Appl. 85 (1987) 17–27], in which they showed that σ is a product of simple elements and gave an upper bound of the smallest number of such factors of σ, whereas in the present paper we will give lower bounds of σ in case that R is a local domain. Moreover we will factorize θσ as a product of symmetries and transvections for some θ the matrix of which is diagonal.