On the relation between torsion submodule and Fitting ideals

摘要

Let R be a commutative ring and let N be a submodule of R-n which consists of columns of a matrix A = (a(ij)) with a(ij) is an element of R for all 1 <= i <= n, j is an element of Lambda, where Lambda is an index set. For every mu = {j1, center dot center dot center dot ,j(q)} subset of Lambda, let I-mu(N) be the ideal generated by subdeterminants of size q of the matrix (a(ij) : 1 <= i <= n,j is an element of mu). Let M = R-n/N. In this paper, we obtain a constructive description of T(M) and we show that when R is a local ring, M/T(M) is free of rank n - q if and only if I-mu(N) is a principal regular ideal, for some mu = {j1, center dot center dot center dot ,j(q)}subset of Lambda. This improves a lemma of Lipman which asserts that, if I(M) is the (m - q)th Fitting ideal of M then I(M) is a regular principal ideal if and only if N is finitely generated free and M/T(M) is free of rank m - q.