Fitting ideal

In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements. They were introduced by .
Definition.
If "M" is a finitely generated module over a commutative ring "R" generated by elements "m"1...,"m""n"
with relations
then the "i"th Fitting ideal Fitt"i"("M") of "M" is generated by the minors (determinants of submatrices) of order "n" − "i" of the matrix "a""jk". 
The Fitting ideals do not depend on the choice of generators and relations of "M".
Some authors defined the Fitting ideal "I"("M") to be the first nonzero Fitting ideal Fitt"i"("M").
Properties.
The Fitting ideals are increasing
If "M" can be generated by "n" elements then Fitt"n"("M") = "R", and if "R" is local the converse holds. We have Fitt0("M") ⊆ Ann("M") (the annihilator of "M"), and Ann("M")Fitt"i"("M") ⊆ Fitt"i"−1("M"), so in particular if "M" can be generated by "n" elements then Ann("M")"n" ⊆ Fitt0("M").
Examples.
If "M" is free of rank "n" then the Fitting ideals Fitt"i"("M") are zero for "i"<"n" and "R" for "i" ≥ "n". 
If "M" is a finite abelian group of order |"M"| (considered as a module over the integers) then the Fitting ideal Fitt0("M") is the ideal (|"M"|).
The Alexander polynomial of a knot is a generator of the Fitting ideal of the first homology of the infinite abelian cover of the knot complement.
Fitting image.
The zeroth Fitting ideal can be used also to give a definition of scheme-theoretic image of morphisms, which behaves well in families. Given a morphism of schemes formula_2, the Fitting image of "f" is defined to be the closed subscheme associated to the sheaf of ideals formula_3, where formula_4 is seen as a formula_5-module via the canonical morphism formula_6.