{
  "id": "enwiki-001296762563",
  "slug": "fitting-ideal",
  "title": "Fitting ideal",
  "qid": "",
  "lead": "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 .",
  "body": "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 a_{j1}m_1+\\cdots + a_{jn}m_n=0\\ (\\text{for }j = 1, 2, \\dots) then the ith Fitting ideal \\operatorname{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 \\operatorname{Fitt}_i(M) . ==Properties== The Fitting ideals are increasing \\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 Fitt 0 (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  ⊆ Fitt 0 (M). ==Examples== If M is free of rank n then the Fitting ideals \\operatorname{Fitt}_i(M) are zero for i |M| (considered as a module over the integers) then the Fitting ideal \\operatorname{Fitt}_0(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. ==References==",
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    "id": "enwiki-001296762563",
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    "title": "Fitting ideal",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/fitting-ideal",
    "issued": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "accessed": {
      "date-parts": [
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          2026,
          11,
          15
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    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001296762563",
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