Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain - Corollaries

Corollaries

This includes the classification of finite-dimensional vector spaces as a special case, where . Since fields have no non-trivial ideals, every finitely generated vector space is free.

Taking yields the fundamental theorem of finitely generated abelian groups.

Let T be a linear operator on a finite-dimensional vector space V over K. Taking, the algebra of polynomials with coefficients in K evaluated at T, yields structure information about T. V can be viewed as a finitely generated module over . The last invariant factor is the minimal polynomial, and the product of invariant factors is the characteristic polynomial. Combined with a standard matrix form for, this yields various canonical forms:

• invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)
• primary decomposition + companion matrix yields primary rational canonical form
• primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)