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 Hans Fitting (1936).
If M is a finitely generated module over a commutative ring R generated by elements m1,...,mn with relations then the ith Fitting ideal
of M is generated by the minors (determinants of submatrices) of order
The Fitting ideals do not depend on the choice of generators and relations of M. Some authors defined the Fitting ideal
to be the first nonzero Fitting ideal
The Fitting ideals are increasing If M can be generated by n elements then Fittn(M) = R, and if R is local the converse holds.
If M is free of rank n then the Fitting ideals
are zero for i (considered as a module over the integers) then the Fitting ideal 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. The zeroth Fitting ideal can be used also to give a variant of the notion of scheme-theoretic image of a morphism, a variant that behaves well in families. Specifically, given a finite morphism of noetherian schemes is coherent, so we may define is called the Fitting image of f.[1][citation needed] This commutative algebra-related article is a stub.