Principal ideal domains and other rings
Over a principal ideal domain \(R\) – for example \(\ZZ\), \(k[x]\) and \(k[\![x]\!]\) for a field \(k\), and every Euclidean domain – the structure theorem classifies finitely generated \(R\)-modules: each is the direct sum of a free module and a torsion module, and the torsion module is determined up to isomorphism by its invariant factors.
Over a ring that is not a PID, such as \(k[x,y]\), \(\ZZ[x]\), or a quotient \(k[x,y]/I\), the structure theorem does not apply. For finitely generated modules over such a ring, the properties of being free, projective, flat, and torsion-free can differ; over a PID they coincide.
The computation over a PID
Let \(R\) be a PID and \(M\) a finitely generated \(R\)-module.
Presentation. Write \(M \cong R^n / \im(A)\) for a matrix \(A\) over \(R\). For an abelian group given by generators and relations, \(A\) is the relation matrix.
Smith normal form. The Smith normal form of \(A\) has diagonal entries \(a_1 \divides a_2 \divides \cdots \divides a_k\), with \(a_i = d_i/d_{i-1}\), where \(d_0=1\) and \(d_i\) is a greatest common divisor of the \(i\times i\) minors of \(A\). See Smith normal form.
Decomposition. \[ M \cong R^{n-k} \oplus \bigoplus_{i=1}^k R/\gens{a_i}. \] The free rank of \(M\) is \(n-k\), and the summands \(R/\gens{a_i}\) with \(a_i\) a unit are zero; the nonunit \(a_i\) are the invariant factors of \(M\).
Elementary divisors and invariant factors. The elementary divisors are the prime-power factors of the invariant factors. Conversely, the largest invariant factor is the product, over the primes \(p\), of the largest elementary divisor that is a power of \(p\); removing those elementary divisors and repeating gives the next invariant factor.
Three instances of the structure theorem
| Ring | Module | Name of the classification |
|---|---|---|
| \(\ZZ\) | abelian group | structure theorem for finitely generated abelian groups |
| \(k[x]\) | vector space with a linear operator | rational canonical form |
| \(k[x]\), with the characteristic polynomial split over \(k\) | vector space with a linear operator | Jordan canonical form |
A square matrix \(A\) over a field \(k\) makes \(k^n\) a \(k[x]\)-module with \(x\) acting by \(A\), and the invariant factors of this module are the invariant factors of \(xI-A\).
Free, projective, flat, and torsion-free modules
For modules over an integral domain, \[ \text{free} \implies \text{projective} \implies \text{flat} \implies \text{torsion-free}, \] and none of the implications reverses in general. For finitely generated modules over a PID, the four conditions coincide.
(Separating the conditions)
\(\ZZ/2\) over \(\ZZ/6\) is projective and not free: \(\ZZ/6\cong\ZZ/2\times\ZZ/3\), so \(\ZZ/2\) is a direct summand of a free module, and it has \(2<6\) elements.
\(\QQ\) over \(\ZZ\) is flat, not projective, and not finitely generated.
The ideal \(\gens{2,x} \subseteq \ZZ[x]\) is torsion-free and not free.
Free ideals
A nonzero ideal \(I\) of an integral domain \(R\) is free as an \(R\)-module if and only if it is principal (An ideal of an integral domain is free if and only if it is principal). For distinct nonzero \(m_1,m_2\in I\), the relation \(m_2\cdot m_1 - m_1\cdot m_2 = 0\) has nonzero coefficients, so a basis of \(I\) has one element. In particular, every nonprincipal ideal of an integral domain is a torsion-free module that is not free.