Finitely presented module ($R^m⟶R^n⟶M⟶0$)

A module is finitely presented if it has finitely many generators and finitely many defining relations, equivalently an exact sequence of the displayed form with m and n finite. Over a commutative ring a relation matrix records the presentation, and its minors define the Fitting ideals.

// Target: knot-theory.bigb

For a module over a commutative ring presented by a matrix with n generators, its zeroth Fitting ideal is generated by all n-by-n minors of the relation matrix. It is independent of the chosen finite presentation. Over a unique factorization domain, the greatest common divisor of these minors defines the module order up to a unit. This does not require a Smith normal form over the ring. The square Seifert-matrix presentation of the Alexander module makes the order its determinant.

// Target: knot-theory.bigb