If M is a finitely generated module over a commutative ring R generated by elements m₁,...,m_n with relations

${\displaystyle a_{j1}m_{1}+\cdots +a_{jn}m_{n}=0\ ({\text{for }}j=1,2,\dots )}$ 

then the ith Fitting ideal ${\displaystyle \operatorname {Fitt} _{i}(M)}$  of M is generated by the minors (determinants of submatrices) of order ${\displaystyle n-i}$  of the matrix ${\displaystyle a_{jk}}$ . The Fitting ideals do not depend on the choice of generators and relations of M.

Some authors defined the Fitting ideal ${\displaystyle I(M)}$  to be the first nonzero Fitting ideal ${\displaystyle \operatorname {Fitt} _{i}(M)}$ .

The Fitting ideals are increasing

${\displaystyle \operatorname {Fitt} _{0}(M)\subseteq \operatorname {Fitt} _{1}(M)\subseteq \operatorname {Fitt} _{2}(M)\subseteq \cdots }$ 

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).

If M is free of rank n then the Fitting ideals ${\displaystyle \operatorname {Fitt} _{i}(M)}$  are zero for i<n and R for i ≥ n.

If M is a finite abelian group of order ${\displaystyle |M|}$  (considered as a module over the integers) then the Fitting ideal ${\displaystyle \operatorname {Fitt} _{0}(M)}$  is the ideal ${\displaystyle (|M|)}$ .

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.

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 ${\displaystyle f\colon X\rightarrow Y}$ , the ${\displaystyle {\mathcal {O}}_{Y}}$ -module ${\displaystyle f_{*}{\mathcal {O}}_{X}}$  is coherent, so we may define ${\displaystyle \operatorname {Fitt} _{0}(f_{*}{\mathcal {O}}_{X})}$  as a coherent sheaf of ${\displaystyle {\mathcal {O}}_{Y}}$ -ideals; the corresponding closed subscheme of ${\displaystyle Y}$  is called the Fitting image of f.^{[1][citation needed]}

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• Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae, 76 (2): 179–330, doi:10.1007/BF01388599, ISSN 0020-9910, MR 0742853
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