In this section we will give careful statements of various theorems related to the structure of finitely generated modules over a PID. We will then illustrate these statements in two specific settings: the case when \(R=\Z\text{,}\) where we obtain a classification of finitely generated abelian groups; and the case when \(R=F\) is a field, in which case we obtain a classification of linear transformations of finite-dimensional vector spaces in terms of their rational canonical form.
A torsion element of \(M\) is an element \(m\in M\) such that \(rm=0\) for some nonzero \(r\in R\text{.}\) The set of all torsion elements of \(M\) is denoted \(\Tor M\text{.}\) The module \(M\) is called torsion if \(\Tor M=M\text{,}\) and torsion free if \(\Tor M=\{0\}\text{.}\)
The decomposition in (2.19.4) is unique in the following sense. Given any other decomposition \(M\cong R^{n'}\oplus\bigoplus_{i=1}^{m'}R/(b_i)\text{,}\) where \(b_i\notin R^*\) for all \(i\) and \(b_i\mid b_{i+1}\) for all \(1\leq i\leq m'-1\text{,}\) we have \(n=n'\text{,}\)\(m=m'\text{,}\) and \(a_i\) and \(b_i\) are associates for all \(1\leq i\leq m\text{.}\)
Definition2.19.3.Betti number and invariant factors.
Let \(M\) be a finitely generated module over the PID \(R\text{.}\) A decomposition of \(M\) as in (2.19.4) is called an invariant decomposition. The integer \(n\) appearing in this decomposition is called the betti number of \(M\text{,}\) and the ideals \((a_i)\) are called the invariant factors of \(M\text{.}\) Similarly, the elements \(a_i\) themselves are called invariant factors of \(M\text{.}\)
If \(M\cong R^n\) is free, then \(\Tor M\cong \Tor(R)^n=\{0\}\text{.}\) Conversely, if \(\Tor M=\{0\}\text{,}\) then by (2.19.3) we must have \(\bigoplus_{i=1}^m R/(a_i)=\{0\}\text{,}\) which can only happen if \(m=0\) since the \(a_i\) are nonzero non-units by definition. But then \(M\cong R^n\text{,}\) and is thus free.
Definition2.19.6.\(\alpha\)-torsion and \(\pi\)-primary submodules.
Let \(R\) be an integral domain, and let \(M\) be an \(R\)-module. Given an element \(\alpha\in R\text{,}\) the \(\alpha\)-torsion submodule of \(M(\alpha)\) of \(M\) is defined as
If \(\Tor(M)\ne \{0\}\text{,}\) there exist pairwise nonassociate irreducible elements \(\pi_1,\pi_2,\dots, \pi_r\text{,}\) and for each \(1\leq i\leq r\) positive integers \(n_{i,1}\leq n_{i,2}\leq \dots \leq n_{i,m_i}\) such that
\begin{align}
\Tor M \amp \cong \bigoplus_{i=1}^r M(\pi_i^\infty)\tag{2.19.9}
\end{align}
where the \(\pi_i'\) are pairwise nonassociate irreducible elements and \(n_{i,j}\leq n_{i,j+1}\) for all relevant \(i\) and \(j\text{,}\) we have \(n'=n\text{,}\)\(r'=r\text{,}\)\(m_i'=m_i\) for all \(i\text{,}\)\(n_{i,j}'=n_{i,j}\) for all \(i,j\text{,}\) and \(\pi_i'\) is associate to \(\pi_i\) for all \(i\text{.}\)
In particular, the integers \(n\text{,}\)\(r\text{,}\) and \(n_{i,j}\) are uniquely determined \(M\text{,}\) as are the maximal ideals \((\pi_i)\text{.}\)
Let \(M\) be a finitely generated module over the PID \(R\text{.}\) Given a decomposition of \(M\) as in (2.19.11), the ideals \((\pi_i^{n_{i,j}})\) are called the elementary divisors of \(M\text{.}\) Similarly, the irreducible powers \(\pi_i^{n_{i,j}}\) themselves are called elementary divisors of \(M\text{.}\)
Recall that a \(\Z\)-module is the same thing as an abelian group. Thus, the PID-module structure theorems above give us a classification of finitely generated abelian groups. In a word, it says that any finitely generated abelian group \(A\) can be written as
\begin{align*}
A \amp \cong \Z^n \bigoplus_{i=1}^m \Z/a_i\Z\text{,}
\end{align*}
where the integer \(n\) is uniquely determined by \(A\text{.}\) Furthermore, since in this case the torsion component of \(A\) is a finite direct sum of groups of the form \(\Z/a\Z\) for some positive integer \(a\text{,}\)\(\Tor A\) is finite! This is worth making official.
We need only concern ourselves with the second statement. First note that all elements of \(\Q\Z\) are torsion since \(n(m/n+\Z)=m+\Z=0\in \Q/\Z\text{.}\) It follows that
Note further that \(\Q/\Z\) is infinite: it is easy to see that \(\ord(1/n+\Z)=n\text{,}\) and hence that \(1/n+\Z\ne 1/m+\Z\) for all distinct positive integers \(m\) and \(n\text{.}\) Since \(\Tor(\Q.\Z)=\Q/\Z\) is an infinite group, it follows from the first statement that \(\Q/\Z\) cannot be finitely generated as an abelian group.
Let’s revisit briefly how we used the theory above to classify finite abelian groups. Given such a group \(A\text{,}\) since it is finite, it is finitely generated and \(A=\Tor(A)\text{.}\) It follows that \(A\) can be written as a direct sum of groups of the form \(\Z/a\Z\text{.}\)
When asked to count (or enumerate) the number of distinct abelian groups up to isomorphism of a fixed cardinality \(n\text{,}\) the question naturally arises as to which of the two theorems above we should make use of: Theorem 2.19.2 or Theorem 2.19.7? If we have a irreducible factorization \(n=\prod_{i=1}^r p_i^{n_i}\text{,}\) then the elementary divisor decomposition is a particularly convenient one. Why? A simple counting argument, together with this theorem, tells us that \(\abs{A(p_i^\infty)}=p_i^{n_i}\text{,}\) in which case the possible decompositions of \(A(p_i^{\infty})\) as in Theorem 2.19.7 correspond to partitions of the integer \(n_i\text{:}\) that is, to sequences
Since \(A=A_2\oplus A_3\text{,}\) and there are 3 possibilities for \(A_2\) and \(2\) possibilities for \(A_3\text{,}\) then there are \(3\cdot 2=6\) possibilities for \(A\text{.}\) Thus, up to isomorphism, there are \(6\) distinct abelian groups of cardinality \(72\text{.}\)
