On finitely generated module whose first nonzero Fitting ideal is a product of co-maximal ideals

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] be a finitely generated [Formula: see text]-module. Let [Formula: see text] be the first nonzero Fitting ideal of [Formula: see text]. The main result of this paper is to determine the torsion submodule of modules whose first nonzero Fitting ideals are a product of co-maximal ideals. Also we characterize some modules of this type.